hilbert schemes of points on some classes of surface singularitiesagyenge/thesis_phd.pdf · hilbert...

Hilbert schemes of points on some classesof surface singularities

A Ph.D. dissertation by

Adam Gyenge

submitted to

Eotvos Lorand UniversityInstitute of Mathematics

Doctoral School: Mathematics

Director: Miklos Laczkovich, D.Sc.

Professor, Member of the Hungarian Academy of Sciences

Doctoral Program: Pure Mathematics

Director: Andras Szucs, D.Sc.

Professor, Member of the Hungarian Academy of Sciences

Supervisor: Andras Nemethi, D.Sc.

Professor, Doctor of the Hungarian Academy of Sciences

Consultant: Balazs Szendroi, Ph.D.

Professor, University of Oxford

2016

i

“Don’t Panic.”– Douglas Adams, The Hitchhiker’s Guide to the Galaxy

ii

Acknowledgement

Completing the PhD and writing this thesis was an amazing journey that would

not have been possible without the support and encouragement of many outstanding

people.

My greatest appreciation and gratitude goes to my advisors. Balazs Szendroi

helped a lot in developing the material of the thesis as well as in making it much

more readable than it could be without him. Andras Nemethi pointed out many

mistakes in earlier versions and suggested several improvements. My gratitude

extends to Alastair Craw, Eugene Gorsky and Tamas Szamuely for helpful comments

and discussions about particular problems related to the thesis.

It was a pleasure and a great honour to work at two excellent mathematical

schools in Budapest. The environments of both the ELTE and the Renyi Insitute

were fantastic and motivating. I am grateful for this to Balazs Csikos and Andras

Stipsicz respectively.

Among the unnumerous outstanding teachers during my university years I am

especially thankful to Gabor Etesi, Andras Szucs, Szilard Szabo, Laszlo Feher and

Gyula Lakos.

Moreover, I would like to thank to a variety of young mathematicians who also

helped in several smaller or bigger questions during my work, or with whom we

just had a lot of fun and many interesting conversations. Among many others this

includes Jozsef Bodnar, Gergo Pinter, Levente Nagy, Norbert Pintye, Tamas Laszlo,

Marco Golla, Stefan Behrens and Baldur Sigurdsson.

Finally, I would like to express my deepest gratitude to my family: Mom, Dad,

Zsuzsi, Akos and especially to my beloved Luca and Agi (who had to suffer the most

from the long evenings and weekends while I was finishing the thesis). They all stood

by me and shared with me both the great and the difficult moments of life.

Contents

Chapter 1. Introduction 1

1.1. Problem setting and background motivation 1

1.2. The structure of the thesis 2

Chapter 2. Preliminaries 5

2.1. Hilbert scheme of points and related moduli spaces 5

2.2. Quotient surface singularities and their Hilbert schemes 7

2.3. Quiver variety description of the moduli spaces 10

2.4. Representations of affine Lie algebras 12

Chapter 3. The main results and their corollaries 15

3.1. The main results 15

3.2. The S-duality conjecture 20

Chapter 4. Abelian quotient singularities 23

4.1. Type A basics 23

4.2. Partitions, torus-fixed points and decompositions 23

4.3. First approach: generalized Frobenius partitions 28

4.4. Second approach: abacus configurations 32

4.5. Outlook: cyclic quotient singularities of type (p, 1) 37

Chapter 5. Type Dn: ideals and Young walls 43

5.1. The binary dihedral group 43

5.2. Young wall pattern and Young walls 43

5.3. Decomposition of C[x, y] and the transformed Young wall pattern 45

5.4. Subspaces and operators 46

5.5. Cell decompositions of equivariant Grassmannians 47

5.6. The Young wall associated with a homogeneous ideal 50

Chapter 6. Type Dn: decomposition of the orbifold Hilbert scheme 53

6.1. The decomposition 53

6.2. Incidence varieties 57

6.3. Proof of Theorem 6.3 60

6.4. Digression: join of varieties 64

6.5. Preparation for the proof of the incidence propositions 65

6.6. Proofs of propositions about incidence vareties 73

iii

iv CONTENTS

Chapter 7. Type Dn: special loci 77

7.1. Support blocks 77

7.2. Special loci in orbifold strata and the supporting rules 78

7.3. Special loci in Grassmannians 81

7.4. Proof of Theorem 7.2 83

Chapter 8. Type Dn: decomposition of the coarse Hilbert scheme 85

8.1. Guide to Chapter 8 85

8.2. Distinguished 0-generated Young walls 88

8.3. The decomposition of the coarse Hilbert scheme 91

8.4. Possibly and almost invariant ideals 93

8.5. Euler characteristics of strata and the coarse generating series 96

Chapter 9. Type Dn: abacus combinatorics 103

9.1. Guide to Chapter 9 103

9.2. Young walls and abacus of type Dn 105

9.3. Core Young walls and their abacus representation 106

9.4. 0-generated Young walls and their abacus representations 114

9.5. The generating series of distinguished 0-generated walls 116

Bibliography 123

Summary 127

Osszefoglalo 129

CHAPTER 1

Introduction

This chapter presents the motivations behind the study of Hilbert schemes of

points on surface singularities and sets the aim of the research. Finally the structure

of the thesis is briefly summarized.

1.1. Problem setting and background motivation

The punctual Hilbert scheme parameterizing the zero-dimensional subschemes of

a quasi-projective variety contains a large amount of information about the geometry

and topology of the base variety. Moreover, it is a very important moduli space. As

a set, it consists of ideal sheaves of the sheaf of regular functions on the variety, such

that the quotient by the ideal has finite length. The Hilbert schemes of points on

smooth curves and surfaces have been investigated for a long time by several people,

including Hartshorne [30], Fogarty [16], Macdonald [47], Iarrobino [34], Briancon [5].

Due to the work of Nakajima [52], Grojnowski [23], and many others, it has turned

out that the surface case has an especially rich geometrical structure, see e.g. [53].

The punctual Hilbert scheme of a smooth curve or surface is a smooth variety

partitioned naturally according to the length. For a smooth curve C the expression∑n≥0

χ(Hilbn(C))qn =1

(1− q)χ(C)

for the generating series collecting the Euler characteristics of the different components

is a corollary of the Macdonald formula [47]. Using the Weil conjecture (as proved by

Deligne) Gottsche proved the following remarkable product formula for the generating

series of the Euler characteristics of the Hilbert scheme of points on a smooth surface

S [21]: ∑n≥0

χ(Hilbn(S))qn =∏n≥1

1

(1− qn)χ(S).

In the recent years a new direction has emerged, which also allows singularities

on the base variety. A breakthrough result was obtained by Maulik [48], who proved

the conjecture of Oblomkov and Shende relating an integral of the length function

with respect to the Euler characteristic over the Hilbert scheme of points of a curve

with planar singularities to the HOMFLY polynomial of its link. The result shows

that this polynomial invariant of the link of a singularity contains information about

the invariants of the Hilbert scheme of points on the singularity.

1

2 1. INTRODUCTION

Motivated by these results it is natural to consider the Hilbert scheme of points

on singular surfaces. The aim of this thesis is to describe the Euler characteristics

of the Hilbert schemes parameterizing the zero-dimensional subschemes of of some

basic classes of surface singularities.

The well known simple singularities are the simplest type of normal surface

singularities, and it is known that they have an orbifold structure. There are at

least two natural versions of the punctual Hilbert scheme in the case of quotient

singularities. One of these, the equivariant Hilbert scheme of C2 with respect to

the groups of ADE type were investigated by Nakajima [51]. The other type, the

coarse Hilbert scheme is more mysterious. We show that the generating series of

the Euler characteristics of the coarse Hilbert schemes of points on the singularities

of type An and Dn can be computed from the multivariable generating series of

the corresponding equivariant Hilbert schemes. The remaining cases E6, E7 and E8

are not treated here, but computer calculations lead us to an analogous conjecture.

The proofs might be similar as well, once the representation theoretic tools become

available.

The moduli spaces of torsion free sheaves on surfaces are higher rank analogs of

the Hilbert schemes. In type A our results reveal their Euler characteristic generating

function as well. Another, very interesting class of normal surface singularities is the

so-called cyclic quotient singularities of type (p, 1). As an outlook we also obtain

some results about the associated generating functions.

1.2. The structure of the thesis

The thesis summarizes the results of the papers [26], [27], [28], [29], the last two

of which are joint manuscripts with Andras Nemethi and Balazs Szendroi.

The thesis can be divided into two main parts. The first part contains the

preliminaries, statement of all the major results, and the proofs for the type A and

the type (p, 1) cyclic quotient cases.

• Chapter 2 contains the preliminaries for the whole thesis. We summarize

the necessary background and introduce the two types of Hilbert schemes

which will be our central objects. The quiver description of the moduli space

of higher rank torsion-free sheaves is also summarized, together with the

necessary background on representation theory. Parts of this chapter have

appeared in [29] and [28].

• Chapter 3 announces the major results of the thesis as well as their

corollaries related to the S-duality conjecture. These appear in [26], [29]

and [27].

• In Chapter 4 we first investigate the type A case from several viewpoints.

We give a new proof to Theorem 3.2 following [26]. Second, we establish a

connection between the two types of Hilbert schemes and prove Theorem 3.5.

1.2. THE STRUCTURE OF THE THESIS 3

Individually, both Theorem 3.2 and Theorem 3.5 were already known in the

literature. But the unified treatment using the so-called abacus combinatorics

has the advantage that it generelizes away from type A case. The treatment

of this second path is based on [28]. As an outlook, we prove the results

about the cyclic quotient singularities of type (p, 1) as in [27].

The second part is based entirely on [28], and contains the proofs for the type D

case.

• In Chapter 5, we introduce Schubert-style cell decompositions of Grass-

mannians of homogeneous summands of C[x, y].

• In Chapter 6 we give a cell decomposition of the orbifold Hilbert scheme,

proving Theorem 3.2 in type D.

• In Chapter 7, we discuss some special subsets of the strata and their

geometry.

• A decomposition of the coarse Hilbert scheme is given in Chapter 8.

• In Chapter 9, the proof of Theorem 3.5 in type D is completed using

combinatorial enumeration.

CHAPTER 2

Preliminaries

This chapter contains the necessary background for the thesis. We introduce the

Hilbert scheme of points and its higher rank analog in the equivariant setting with

respect to a finite group action. We mention their connection to quiver varieties and

to the representation theory of Lie algebras of affine type.

2.1. Hilbert scheme of points and related moduli spaces

For a general reference on Hilbert scheme of points on surfaces, we refer the

reader to [53]. Here we recall the basic facts.

Let X be a projective scheme over an algebraically closed field k and OX(1) a

very ample line bundle on X. The contravariant Hilbert functor

Hilb(X) : [Schemes]→ [Sets]

from the category of schemes to the category of sets is defined as

Hilb(X)(U) = {Z ⊂ X×U : Z is a closed subscheme, π : Z ↪→ X×U → U is flat}.

This means that the functor Hilb(X) associates to a scheme U the set of families of

closed subschemes on X which are parameterized by U .

For u ∈ U , the Hilbert polynomial in u is defined as

Pu(k) = χ(OZu ⊗OX(k)),

where Zu = π−1(u). By the flatness of Z over U , Pu is constant on the connected

components of U . For each polynomial P let HilbP (X) be the subfunctor of Hilb(X)

which associates to U the set of families of closed subschemes in X which has P as

its Hilbert polynomial.

Theorem 2.1 ([24]). The functor HilbP (X) is representable by a projective

scheme HilbP (X). In particular, there exists a universal family Z over HilbP (X),

and that every family on U is induced by a unique morphism φ : U → HilbP (X).

If there is an open subscheme Y of X, then there exists the corresponding open

subscheme HilbP (Y ) of HilbP (X) parameterizing subschemes in Y . In particular,

HilbP (Y ) is defined for a quasi-projective scheme Y as well.

5

6 2. PRELIMINARIES

Definition 2.2. For some fixed integer m ∈ Z, let P be the constant Hilbert

polynomial given by P (k) = m for all k ∈ Z. We denote by Hilbm(X) the corre-

sponding Hilbert scheme. It is called the Hilbert scheme of m points on X, or the

punctual Hilbert scheme of X if m is not important or specified previously.

As a set, Hilbm(X) consists of sheaf of ideals I of the structure sheaf OX of X,

such that OX/I is a finite length OX-module with H0(OX/I) an m-dimensional

C-vector space. For a locally closed subvariety Y ⊂ X, let Hilbm(X, Y ) ⊂ Hilbm(X)

be the Hilbert scheme of zero-dimensional subschemes of X of length m supported

set-theoretically at the points of Y . The disjoint union of these spaces for all m is

denoted as Hilb(X), and Hilb(X, Y ) respectively.

Let us fix an arbitrary quasi-projective variety X. A central invariant in this

thesis will be the generating series of the Euler characteristics of its Hilbert schemes:

(2.1) ZX(q) =∞∑m=0

χ (Hilbm(X)) qm.

There is a relation between the Hilbert scheme of m points to the m-th symmetric

product of X given by Hilbert-Chow morphism:

π : Hilbm(X) → SmX

I 7→∑

x∈X colength(Ix)[x].

It is known that if X is a nonsingular curve then Hilbm(X) = SmX. Then we

have MacDonald’s result [47] for the generating series:

ZC(q) = (1− q)−χ(C).

Another well investigated case is when X is a smooth surface. Then the following

theorem holds:

Theorem 2.3 (Fogarty). (1) Hilbm(X) is smooth of dimension 2m.

(2) The morphism π : Hilbm(X)→ SmX is a resolution of singularities.

For any nonsingular surface S, we have (a specialization of) Gottsche’s formula [21]

(2.2) ZS(q) =

(∞∏m=1

(1− qm)−1

)χ(S)

.

For results valid for higher dimensional varieties, see [8].

A particular case of the general construction above is when X = C2. Hilb(C2)

has a rich geometric structure and will play a crucial rule throughout the thesis.

For singular varieties X, the series ZX(q) is much less studied. For a singular

curve C with a finite set of singuliarities {P1, . . . , Pk} , and under the assumption

that (C, pi) is planar for each i, we have the beautiful conjecture of Oblomkov and

2.2. QUOTIENT SURFACE SINGULARITIES AND THEIR HILBERT SCHEMES 7

Shende [57], proved by Maulik [48], which takes the form

(2.3) ZC(q) = (1− q)−χ(C)

k∏j=1

Z(Pi,C)(q).

Here Z(Pi,C)(q) are highly nontrivial local terms that depend only on the embedded

topological type of the link of the singularity Pi ∈ C.

The higher rank analog of the Hilbert scheme of points on C2 is the moduli space

of framed torsion free sheaves on P2. Torsion free sheaves are generalizations of

vector bundles, essentially they can be viewed as vector bundles which are allowed

to have some singularities: the dimensions of the fibers do not all have to be equal.

Their moduli space is defined as

Mr,m(C2) =

(E,Φ)

∣∣∣∣∣∣∣∣∣E is a torsion free sheaf of rank r,

c2(E) = m which is locally free in a

neighbourhood of l∞,

Φ: E|l∞∼−→ O⊕rl∞ is a framing at infinity

/

isomorphism,

where l∞ = {[0 : z1 : z2] ∈ P2} ⊂ P2 is the line at infinity. We put C2 instead of

P2 in the argument to keep the analogy with the Hilbert scheme, but this should

not eventuate any confusion. By the existence of a framing Φ, c1(E) = 0 for each

representative (E,Φ) of an element in Mr,m(C2).

Lemma 2.4. Let H = E|C2 be a torsion free sheaf on C2. Then H is the subsheaf

of a free sheaf in a canonical way.

Proof. The double dual sheaf H∨∨ is reflexive. Consequently, it has depth 2.

By the Auslander-Buchsbaum formula, the projective dimension of H∨∨ is 0, i.e.

H∨∨ is projective. This implies, by the Quillen-Suslin theorem, that H∨∨ is also free.

Since H is torsion free, the canonical map H → H∨∨ is injective. �

Corollary 2.5. Any framed torsion free sheaf (E,Φ) on P2 is a subsheaf of a

locally free sheaf in a canonical way.

Since E itself is locally free on l∞, for r = 1 the correspondence

M1,m(C2)∼−→ Hilbm(P2 \ l∞) = Hilbm(C2)

E 7→ E∨∨/E

gives an isomorphism.

2.2. Quotient surface singularities and their Hilbert schemes

Let G < GL(2,C) be a small finite subgroup and denote by C2/G the correspond-

ing quotient variety. There are two different types of Hilbert schemes attached to

this data. First, there is the classical Hilbert scheme Hilb(C2/G) of the quotient

8 2. PRELIMINARIES

space. This is the moduli space of ideal sheaves in OC2/G(C2/G) of finite colength.

We call this the coarse Hilbert scheme of points. It decomposes

Hilb(C2/G) =⊔m∈N

Hilbm(C2/G)

into components which are quasiprojective but singular varieties indexed by “the

number of points”, the codimension m of the ideal. Second, there is the moduli space

of G-invariant finite colength subschemes of C2, the invariant part of Hilb(C2) under

the lifted action of G. This Hilbert scheme is also well known and is variously called

the orbifold Hilbert scheme [64] or equivariant Hilbert scheme [25]. We denote it by

Hilb([C2/G]). This space also decomposes as

Hilb([C2/G]) =⊔

ρ∈Rep(G)

Hilbρ([C2/G]),

where

Hilbρ([C2/G]) = {I ∈ Hilb(C2)G : H0(OC2/I) 'G ρ}

for any finite-dimensional representation ρ ∈ Rep(G) of G; here Hilb(C2)G is the set

of G-invariant ideals of C[x, y], and 'G means G-equivariant isomorphism. Being

components of the fixed point set of a finite group acting on smooth quasiprojective

varieties, the orbifold Hilbert schemes themselves are smooth and quasiprojective [7].

There is a natural pushforward map between the two kinds of Hilbert schemes:

each J ∈ Hilb([C2/G]) can be mapped to its G-invariant part, giving a morphism [6,

3.4]

p∗ : Hilb([C2/G]) → Hilb(C2/G)

J 7→ JG = J ∩ C[x, y]G

called the quotient-scheme map. There is also a set-theoretic pullback map, which

however does not preserve flatness in families, so it is not a morphism between the

Hilbert schemes: the inclusion i : C[x, y]G ⊂ C[x, y] induces a pullback map on the

ideals, and its image is contained in the set of G-equivariant ideals, leading to a map

of setsi∗ : Hilb(C2/G)(C) → Hilb([C2/G])(C)

I 7→ i∗I = C[x, y].I

Since for I � C[x, y]G, we clearly have (C[x, y].I)G = I, the composite p∗ ◦ i∗ is the

identity on the set of ideals of the invariant ring.

We collect the topological Euler characteristics of the two versions of the Hilbert

scheme into two generating functions. Let ρ0, . . . , ρn ∈ Rep(G) denote the (isomor-

phism classes of) irreducible representations of G, with ρ0 the trivial representation.

Definition 2.6. (a) The orbifold generating series of the orbifold [C2/G] is

Z[C2/G](q0, . . . , qn) =∞∑

m0,...,mn=0

χ(Hilbm0ρ0+...+mnρn([C2/G])

)qm0

0 · . . . · qmnn .

2.2. QUOTIENT SURFACE SINGULARITIES AND THEIR HILBERT SCHEMES 9

(b) The coarse generating series of the singularity C2/G is just the series defined in

(2.1):

ZC2/G(q) =∞∑m=0

χ(Hilbm(C2/G)

)qm.

Assume that G < SL(2,C). Then G fixes the line l∞ ⊂ P2 introduced above. Let

us take and fix a lift of the G-action to O⊕rl∞ . This can be written as W ⊗C O⊕rl∞ ,

where W is a representation of G. Then the G-action on C2 lifts naturally to a

G-action on Mr,m(C2) [55]. We define the moduli space of G-equivariant torsion

free sheaves on P2 with a framing as the G-invariant part of the space Mr,m(C2)

with respect to the induced G-action. It will be denoted by Mr,m([C2/G]). The

dependence on the choice of W is suppressed in the notation. In fact, it is easy to

see that the isomorphism type of Mr,m([C2/G]) only depends on the isomorphism

class of W . We denote

Mr(C2) =⊔m

Mr,m(C2).

Using Beilinson’s spectral sequence it can be shown that the analog of the

quotient C[x, y]/I for a higher rank framed sheaf (E, φ) is H1(E(−1)), and also that

H0(E(−1)) = H2(E(−1)) = 0 [53, Chapter 2]. If E is G-invariant, then H1(E(−1))

carries naturally a G-representation, and there is a decomposition

(2.4) Mr([C2/G]) =⊔

ρ∈Rep(G)

Mr,ρ([C2/G]),

where

Mr,ρ([C2/G]) = {(E,Φ) ∈Mr(C2)G : H1(E(−1)) 'G ρ}.

It is possible to define the higher rank analogue of the coarse Hilbert scheme

Hilb(C2/G) and a morphism pG∗ from Mr([C2/G]) to this new moduli space which

“descends” the sheaves from [C2/G] to C2/G when W is the trivial r dimensional

representation of G. The result of this descent map is a higher rank torsion free sheaf

on the variety P2/G which is trivial on l∞/G. The rank one case of this moduli space

coincides with Hilb(C2/G). However, the general behaviour and the computations

seems to be more complicated in the higher rank case even in type A. Therefore,

we leave the investigation of the moduli spaces of higher rank framed torsion free

sheaves on quotient singularities for further study.

The Euler characteristics of the higher rank equivariant moduli spaces for a fixed

isomorphism class of W are collected again into generating series:

Definition 2.7.

ZW[C2/G](q0, . . . , qn) =

∞∑m0,...,mn=0

χ(Mr,m0ρ0+...+mnρn([C2/G])

)qm0

0 · . . . · qmnn .

10 2. PRELIMINARIES

In this thesis almost always we are only concerned with finite subgroups G <

SL(2,C). As it is well known, these are classified into three types: type An for n ≥ 1,

type Dn for n ≥ 4 and type En for n = 6, 7, 8. The type of the singularity can be

parameterized by a simply-laced irreducible Dynkin diagram with n nodes, arising

from an irreducible simply laced root system ∆. These are the following.

Type An: Type Dn:

Type E6: Type E7:

Type E8:

We denote the corresponding group by G∆ < SL(2,C); all other data corre-

sponding to the chosen type will also be labelled by the subscript ∆. Irreducible

representations ρ0, . . . , ρn of G∆ are then labeled by vertices of the affine Dynkin

diagram associated with ∆, which are recalled in 2.3 below. The singularity C2/G∆

is known as a simple (Kleinian, surface) singularity; we will refer to the corresponding

orbifold [C2/G∆] as the simple singularity orbifold. Except when noted, we always

work with this class of singularities.

Additionally, we will also make some investigation with another class of singular-

ities. Fix a positive integer p. Let Zp be the cyclic group of order p with generator g

and let it act on C2 as: g.x = e2πip x, g.y = e

2πiqp y where q is coprime to p. Then we

get an action of Zp on C2 which is free away from the origin. Let X(p, q) denote the

quotient variety. It is called the cyclic quotient singularity of type (p, q).

The Hilbert scheme Hilb(X(p, q)) of points on X(p, q) is the moduli space of

ideals sheaves in OX(p,q)(X(p, q)) of finite colength. The case when q = p− 1 is just

the type A singularity introduced above. We will present results about the other

extreme case when q = 1, that is, about

ZX(p,1)(q) =∞∑m=0

χ (Hilbm(X(p, 1))) qm.

2.3. Quiver variety description of the moduli spaces

Let (I,H) be a quiver. More precisely, I is a set of vertices and H is a set of

oriented edges. Let H = H ∪H∗, where H∗ is the set of edges in H with the reversed

2.3. QUIVER VARIETY DESCRIPTION OF THE MODULI SPACES 11

orientation. For dimension vectors v, w ∈ ZI≥0 we define a Nakajima quiver variety

as follows. See [51, 55] and references therein for the details, here we follow the

notations of [59].

Fix I-graded vector spaces V,W such that dimVi = vi, dimWi = wi. Let

M(v, w) =

⊕h∈H

Hom(Vs(h), Vt(h))

⊕(⊕i∈I

Hom(Wi, Vi)⊕ Hom(Vi,Wi)

),

where h ∈ H is an oriented edge from s(h) to t(h). Note that GL(V ) =∏GL(Vi)

acts on M(v, w) by

(gi) · (Bh, ai, bi) = (gt(h)Bhg−1s(h), giai, big

−1i )

for any gi ∈ GL(Vi), Bh ∈ Hom(Vs(h), Vt(h)), ai ∈ Hom(Wi, Vi) and bi ∈ Hom(Vi,Wi).

Elements in M(v, w) can be shortly denoted as a triple (B, a, b). Here B is a

representation of the path algebra of the quiver on V , while a : W → V and

b : V → W are maps of I-graded vector spaces.

The moment map µ for the GL(V )-action on M(v, w) is given by

µ(B, a, b) =⊕i∈I

∑h:t(h)=i

ε(h)BhBh∗ + aibi

∈⊕i∈I

gl(Vi) = gl(V ) ,

where ε(h) = 1 and ε(h∗) = −1 for h ∈ H. A triple (B, a, b) ∈ M(v, w) is called

stable if im(a) generates V under the action of B. The subset of stable triples in

M(v, w) is denoted as M(v, w)st.

The quiver variety associated to the dimension vectors v, w is

M(v, w) = {(B, a, b) ∈M(v, w)st | µ(B, a, b) = 0}/GL(V ) .

This is well defined only up to a (non-canonical) isomorphism, but since we are only

interested in its topological properties we do not consider its dependence on the

vector spaces V and W here.

Let us fix an irreducible simply laced root system ∆ of rank n. Let (I,H) be the

associated affine Dynkin quiver. The orientation of the edges in H are specified as

in the following diagrams for each type.

Type A(1)n :

1 2 n− 1 n.

0

Type D(1)n :

2 3 n− 3n− 2

0

1

n− 1

n

12 2. PRELIMINARIES

Type E(1)6 :

4 3 2 5 6

1

0

Type E(1)7 :

1 2 3 4 5

7

0 6

Type E(1)8 :

2 3 4 5 6

8

1 70

Recall the moduli space Mr,m([C2/G∆]) of torsion free G∆-equivariant sheaves

with a framing and its decomposition (2.4). All these depended on the isomorphism

class of the G∆-representation W , which can be written as ρ⊕w00 ⊕ · · · ⊕ ρ⊕wnn where

ρ0, . . . , ρn are the irreducible representations of G. Let w = (w0, . . . , wn).

Theorem 2.8. [55, Section 3] Let ρ = ρ⊕v00 ⊕· · ·⊕ ρ⊕vnn be the isomorphism class

of a G∆-representation V , and let v = (v0, . . . , vn). The moduli space Mr,ρ([C2/G∆])

is isomorphic to the quiver variety M(v, w) associated to (I,H) as above and to the

dimension vectors v and w. Therefore, Mr,m([C2/G∆]) has the decomposition

Mr,m([C2/G]) =⊔

v:|v|=m

M(v, w).

In the case of the Hilbert scheme the representation type of W is necessarily the

trivial representation, i.e. w = (1, 0, . . . , 0).

Corollary 2.9. Let ρ = ρ⊕v00 ⊕ · · · ⊕ ρ⊕vnn be the isomorphism class of a G∆-

representation V , and let v = (v0, . . . , vn). The Hilbert scheme Hilbρ([C2/G∆])

is isomorphic to the quiver variety M(v, w) associated to (I,H) as above and to

the dimension vectors v and w = (1, 0, . . . , 0). Therefore, Hilbm([C2/G∆]) has the

decomposition

Hilbm([C2/G∆]) =⊔

v:|v|=m

M(v, w).

2.4. Representations of affine Lie algebras

The author learned the material in this section from Balazs Szendroi.

Let ∆ be an irreducible finite-dimensional root system, corresponding to a complex

finite dimensional simple Lie algebra g of rank n. Attached to ∆ is also an (untwisted)

affine Lie algebra g, but a slight variant will be more interesting for us, see e.g. [15,

Sect 6]. Consider the Lie algebra g⊕ C spanned by elements xtm for x ∈ g, m ∈ Z;

elements aj for j ∈ Z \ {0}, a scaling element d, and a central element c. It is the

2.4. REPRESENTATIONS OF AFFINE LIE ALGEBRAS 13

direct sum of the affine Lie algebra g and an infinite Heisenberg algebra heis, with

their centers identified.

Let V0 be the basic representation of g, the level-1 representation with highest

weight ω0. Let F be the standard Fock space representation of heis, having central

charge 1. Then V = V0 ⊗ F is a representation of g⊕ C that we may call the

extended basic representation. By the Frenkel–Kac theorem [17],

V ∼= Fn+1 ⊗ C[Q∆],

where Q∆ is the root lattice corresponding to the root system ∆. Here, for β ∈ C[Q∆],

Fn+1 ⊗ eβ is the sum of weight subspaces of weight ω0 −(m+ 〈β,β〉

2

)δ + β, m ≥ 0,

with δ being the imaginary root [36]. Thus, we can write the character of this

representation as

(2.5) charV (q0, . . . , qn) = eω0

(∏m>0

(1− qm)−1

)n+1

·∑β∈Q∆

qβ1

1 · · · · · qβnn (q1/2)〈β,β〉,

where q = e−δ, and β = (β1, . . . , βn) is the expression of an element of the finite type

root lattice in terms of the simple roots.

Example 2.10. For ∆ of type An, we have g = sln+1, g = sln+1, g⊕ C = gln+1.

In this case there is in fact a natural vector space isomorphism V ∼= F with Fock

space itself, see e.g. [61, Section 3E].

At least for simply-laced classical types An and Dn (and also for some others, see

[39]), the above representation can be constructed on a vector space spanned by an

explicit (affine) “crystal” basis. In type A, this construction is very well known [49];

the type D construction is more recent [37, 38]. For completeness, we recall the

definition from [41].

Let g be an affine Kac-Moody algebra, {hi}i∈I be the set of simple coroots,

{αi}i∈I the set of simple roots, and P the weight lattice.

Definition 2.11. An (affine) crystal basis for g is a set B together with maps

wt: B → P , εi, ϕi : B → Z ∪ {−∞}, and ei, fi : B → B ∪ {0} for i ∈ I, satisfying

the following conditions; for i ∈ I, and b, b′ ∈ B(1) ϕi(b) = εi(b) + wt(b)(hi),

(2) εi(eib) = εi(b)− 1, ϕi(eib) = ϕi(b) + 1, and wt(eib) = wt(b) + αi, if eib ∈ B,

(3) εi(fib) = εi(b) + 1, ϕi(fib) = ϕi(b)− 1, and wt(eib) = wt(b)− αi, if eib ∈ B,

(4) fib = b′ if and only if b = eib′,

(5) if ϕi(b) = −∞, then eib = fib = 0.

In [32] for many types of affine Lie algebras crystal bases are constructed on

certain combinatorial objects called proper Young walls. In type A and D we will

describe these in detail in the later chapters. For ∆ of type A or D the set of

14 2. PRELIMINARIES

proper Young walls will be denoted as Z∆. To be more precise, the Young wall

pattern for An, (respectively, Dn) introduced in Chapter 4 (respectively, in Chapter 5)

corresponds to the dominant integral weight Λ = Λ0 representation of the affine Lie

algebra g of type An (resp., Dn).

In both type A and D there is a certain subset Y∆ ⊂ Z∆ of Young walls which

are called reduced. These satisfy further combinatorial properties which we omit

here, see [38] for the precise definitions. It is shown in [43] for type An, and in [41]

for type Dn, that there is a bijection

Y∆ ←→ C∆ × Pn .

Here C∆ is the set of core Young wall as in the main part of the text.

By [49, 37] the basic representation V0 of g can be constructed on Y∆, and it is

canonically embedded into the extented basic representation of g⊕ C constructed

on Z∆. The embedding is induced by the inclusion Y∆ ⊂ Z∆. The character of the

basic representation is given by the Weyl-Kac character formula

charV0(q0, . . . , qn) = eω0

(∏m>0

(1− qm)−1

)n

·∑β∈Q∆

qβ1

1 · · · · · qβnn (q1/2)〈β,β〉 .

As before, let Γ < SL(2,C) be a finite subgroup and let ∆ ⊂ ∆ be the corre-

sponding finite and affine Dynkin diagrams. By Corollary 2.9 the equivariant Hilbert

schemes Hilbρ([C2/Γ]) for all finite dimensional representations ρ of GΓ are Nakajima

quiver varieties [51] associated to ∆, with dimension vector determined by ρ, and a

specific stability condition. Ssee [18, 50] for more details for type A.

Nakajima’s general results on the relation between the cohomology of quiver

varities and Kac-Moody algebras, specialized to this case, imply

Theorem 2.12 ([51, 55]). The direct sum of all cohomology groups

H∗(Hilbρ([C2/G])) is graded isomorphic to the extended basic representation V of

the corresponding extended affine Lie algebra g⊕ C defined above.

CHAPTER 3

The main results and their corollaries

This chapter announces the main results of the thesis. As a corollary, a modularity

result for the partition function ZX(q) is proved for certain singular surfaces X,

extending the S-duality type result from the nonsingular case.

3.1. The main results

By 2.3 the Hilbert schemes on [C2/G∆] where ∆ is an irreducible simply-laced

root system have a description as Nakajima quiver varieties. By [54, Section 7], these

quiver varieties have no odd cohomology. Taking into account Theorem 2.12, the

character formula (2.5) implies the following theorem.

Theorem 3.1 ([55]). Let [C2/G∆] be a simple singularity orbifold, where ∆ is

any of the types A, D, and E. Then its orbifold generating series can be expressed as

(3.1)

Z[C2/G∆](q0, . . . , qn) =

(∞∏m=1

(1− qm)−1

)n+1

·∑

m=(m1,...,mn)∈Znqm1

1 ·· · ··qmnn (q1/2)m>·C∆·m,

where q =∏n

i=0 qdii with di = dim ρi, and C∆ is the finite type Cartan matrix

corresponding to ∆.

Our first main result is a strengthening of this theorem. Given a Dynkin diagram

∆ of type A or D, we will recall below in 4.2, respectively 5.2, the definition of a

certain combinatorial set, the set of Young walls Z∆ of type ∆.

Theorem 3.2. Let [C2/G∆] be a simple singularity orbifold, where ∆ is of type

An for n ≥ 1 or Dn for n ≥ 4. Then there exists a decomposition

Hilb([C2/G∆]) =⊔

Y ∈Z∆

Hilb([C2/G∆])Y

into locally closed strata indexed by the set of Young walls Z∆ of the appropriate

type. Each stratum is isomorphic to an affine space of a certain dimension, and in

particular has Euler characteristic χ(Hilb([C2/G∆])Y ) = 1.

For type A, the set of Young walls is simply the set of finite partitions, represented

as Young diagrams, equipped with a diagonal labelling. In this case, Theorem 3.2

is well known; the decomposition in type A is not unique, but depends on a choice

of a one-dimensional subtorus of the full torus (C∗)2 acting on the affine plane C2.

For completeness, we summarize the details in Chapter 4. On the other hand, the

15

16 3. THE MAIN RESULTS AND THEIR COROLLARIES

type D case appears to be new; in this case, our decomposition is unique, there is no

further choice to make.

Remark 3.3. As it was explained in 2.3, the orbifold Hilbert schemes of points for

G < SL(2,C) are Nakajima quiver varieties for the corresponding affine quiver. As it

was shown in [60], certain Lagrangian subvarieties in Nakajima quiver varieties are

isomorphic to quiver Grassmannians for the preprojective algebra of the same type,

parameterizing submodules of certain fixed modules. On the other hand, results of

the recent papers [45, 46] imply that every quiver Grassmannian of a representation

of a quiver of affine type D has a decomposition into affine spaces. The relation

between this decomposition and ours deserves further investigation.

We will explain combinatorially in 4.4, respectively 9.3, that the right hand side

of (3.1) enumerates the set of Young walls Z∆ of the appropriate type. The same

combinatorics appears in the representation theoretic considerations in Section 2.4.

Thus Theorem 3.2 implies Theorem 3.1.

Remark 3.4. In type A, it is easy to refine formula (3.1) to a formula involving

the Betti numbers [18], or the motives [25], of the orbifold Hilbert schemes. We leave

the study of such a refinement in type D to future work; compare to Remark 6.4.

The second main result of the thesis is the following formula, which says that the

coarse generating series is a very particular specialization of the orbifold one.

Theorem 3.5. Let C2/G∆ be a simple singularity, where ∆ is of type An for

n ≥ 1 or Dn for n ≥ 4. Let h∨ be the (dual) Coxeter number of the corresponding

finite root system (one less than the dimension of the corresponding simple Lie algebra

divided by n). Then

ZC2/G∆(q) =

(∞∏m=1

(1− qm)−1

)n+1

·∑

m=(m1,...,mn)∈Znζm1+m2+···+mn(q1/2)m

>·C∆·m,

where ζ = exp(

2πi1+h∨

)and C∆ is the finite type Cartan matrix corresponding to ∆.

Thus ZC2/G∆(q) is obtained from Z[C2/G∆](q0, . . . , qn) by the substitutions

q1 = · · · = qn = exp

(2πi

1 + h∨

), q0 = q exp

(− 2πi

1 + h∨

∑i 6=0

dimρi

).

Remark 3.6. The single variable generating series ZC2/G∆in type A was cal-

culated by Toda in [62] using threefold machinery including a flop formula for

Donaldson–Thomas invariants of certain Calabi–Yau threefolds. He does not mention

any connection to Lie theory. The combinatorics, and the one-variable formula for

Z0∆(q), were already known to Dijkgraaf and Su lkowski [12]. They do not give the

interpretation of the combinatorial formula in terms of Hilbert schemes, though

3.1. THE MAIN RESULTS 17

they are clearly motivated by closely related ideas. Instead, they investigate the

partition functions of certain supersymmetric field theories on the so-called ALE

spaces (see [11, 9] for further developments in this direction). The method in [12] for

the proof of Theorem 3.5 in type A is different, using the method of Andrews [2].

This has motivated a new proof of us for Theorem 3.2 in type A appearing in [26],

which we explain here in Section 4.3. Our other main contribution is the general

Lie-theoretic formulation, as well as a proof in type D described in Chapters 5-9 as

in [28]. Along the way, we also provide another direct combinatorial proof in type

A as well, which appears to be new. We believe that already in type A our new

proof is preferable since it directly exhibits the clear connection between the orbifold

and coarse generating series. Also, as we show in the second part of the thesis, this

method generalizes away from type A.

One can check directly that the generating series in Theorem 3.5 has also integer

coefficients for E6, E7 and E8 to a high power in q. This motivates the following.

Conjecture 3.7. Let C2/G∆ be a simple singularity of type En for n = 6, 7, 8.

Let h∨ be the (dual) Coxeter number of the corresponding finite root system. Then,

as for other types,

ZC2/G∆(q) =

(∞∏m=1

(1− qm)−1

)n+1

·∑

m=(m1,...,mn)∈Znζm1+m2+···+mn(q1/2)m

>·C∆·m,

where ζ = exp(

2πi1+h∨

)and C∆ is the finite type Cartan matrix corresponding to ∆.

The key tool in our proof of Theorem 3.5 for types A and D is the combinatorics

of Young walls, in particular their abacus representation. We are not aware of such

explicit combinatorics in type E. We hope to return to this question in later work.

Remark 3.8. We are dealing here with Hilbert schemes, parameterizing rank

r = 1 sheaves on the orbifold or singular surface. In the relationship between the

instantons on algebraic surfaces and affine Lie algebras, level equals rank [23]. Indeed

the (extended) basic represenatation underlying the Young wall combinatorics (see

Appendix) has level l = 1. Thus the substitution above is by the root of unity

ζ = exp(

2πil+h∨

), with l = 1 and h∨ the (dual) Coxeter number. There is an intriguing

analogy here with the Verlinde formula, which uses a similar substitution, into

characters of Lie algebras, by a root of unity ζ = exp( 2πil+h∨

), where l again is the

level, and h∨ the (dual) Coxeter number of the root system of the Lie algebra of the

gauge group. The geometric significance of this observation, if any, is left for future

research.

In the type A case we can go further to the higher rank case. The higher rank

orbifold generating series can again be derived from the results of [55] or [18], but

our methods give a new proof in this case as well. Let W be the isomorphism class


of the framing and let a = (0, . . . , 0, . . . , n− 1, . . . , n− 1), where for each c ∈ C the

number of c’s in a is wc.

Theorem 3.9. Let [C2/G∆] be a simple singularity orbifold, where ∆ is of type

An for n ≥ 1. Then

ZW[C2/G∆](q) =

l∏m=1

Z∆(am)(q),

where l is the length of a,

Z∆(a)(q) =

(∞∏m=1

(1− qm)−1

)n+1

·∑

m=(m1,...,mn)∈Znqm1

1+a · · · · · qmnn+a(q1/2)m

>·C·m ,

q =∏n

i=0 qi, and C is the Cartan matrix of finite type An.

Let us consider now the cyclic quotient singularities introduced in the end of

Section 2.2. Our main result in this direction is a representation of ZX(p,1)(q) as

coefficient of a two variable generating function. In this two variable generating

function continued fractions appear. We introduce the notation [z0]∑

nAnzn = A0.

Theorem 3.10. Let F (q, z) and T (q, z) be the functions defined in (4.17) and

(4.20) below, respectively. Then:

ZX(p,1)(q) = [z0]T (q, z)(F (q−1, z−1)− (qz)−pF (q−1, (qz)−1)

).

Remark 3.11. (1) For p = 1, X(1, 1) = C2. Then, by Theorem 3.5,

ZX(1,1)(q) is also equal to

∞∏m=1

1

1− qm.

(2) For p = 2, X(2, 1) is the A1 singularity. By Theorem 3.5, ZX(2,1)(q) is also

equal to (∞∏m=1

(1− qm)−1

)2

·∑m∈Z

ξmqm2

,

where ξ = exp(2πi3

).

We are not aware of a direct proof for these equalities.

Finally, we are able to globalize the statements so far. Let X = S be a quasi-

projective surface which is non-singular outside a finite number of simple surface

singularities {P1, . . . , Pk}, with (Pi ∈ S) a singularity locally analytically isomorphic

to (0 ∈ C2/G∆i) for G∆i

< SL(2,C) a small finite subgroup, or to (0 ∈ (X(pi, 1)))

for a positive integer pi. Let S0 ⊂ S be the nonsingular part of S.

3.1. THE MAIN RESULTS 19

Theorem 3.12. The generating function ZS(q) of the Euler characteristics of

Hilbert schemes of points of S has a product decomposition

(3.2) ZS(q) =

(∞∏m=1

(1− qm)−1

)χ(S0)

·k∏j=1

Z(Pi,S)(q).

The local terms can be expressed

• either as

(3.3) Z(Pi,S)(q) = ZC2/G∆i(q)

if (Pi ∈ S) ∼= (0 ∈ C2/G∆i), and they are given by Theorem 3.5 for type A

and D, and, assuming Conjecture 3.7, also for type E;

• or as

(3.4) Z(Pi,S)(q) = ZX(pi,1)(q)

if (Pi ∈ S) ∼= (0 ∈ (X(pi, 1))), and they are given by Theorem 3.10.

Proof. The product decomposition (3.2), as well as the equalities (3.3)-(3.4),

follow from a standard argument; we sketch the details. For a point P ∈ S on a

quasiprojective surface S, let HilbmP (S) denote the punctual Hilbert scheme of S at

P , the Hilbert scheme of length m subschemes of S set-theoretically supported at

the single point P . Then for our surface S and for a, let’s say, simple singularity Piwe have

χ(HilbmPi(S)) = χ(Hilbm0 (C2/G∆i)) = χ(Hilbm(C2/G∆i

)).

Here the first equality follows from the analytic isomorphism between (Pi ∈ S) and

(0 ∈ C2/G∆i). The second equality follows from torus localization, using the fact

that each singularity C2/G∆iis weighted homogeneous, admitting a C∗-action. Any

finite length quotient whose set-theoretic support is not equal to 0 is a member of a

nontrivial orbit of this action. Therefore,

χ(Hilbm(C2/G∆i) \ Hilbm0 (C2/G∆i

)) = 0,

and we have the decomposition

Hilbm(S) =⊔

∑ki=0 mi=m

Hilbm0(S0)×k∏i=1

HilbmiPi (S).

Reinterpreting this equality for generating series proves (3.2)-(3.4), recalling also (2.2)

for S0. �

Formulas (3.2)-(3.3) are our analogue for the case of surfaces with simple singu-

larities of the Oblomkov–Shende–Maulik formula (2.3). Note that each C2/G∆iis in

particular a hypersurface singularity, as are planar singularities in the curve case.

The main difference with formula (2.3) is the fact that (conjecturally, for type E)

our local terms Z(Pi,S)(q) are expressed in terms of Lie-theoretic and not topological


data. We leave the question whether our local terms have any interpretation in terms

of the topology of the (embedded) link of Pi, and whether there are nice formulas

for other two-dimensional hypersurface singularities, for further work.

3.2. The S-duality conjecture

We follow here [22]. Let S be a smooth projective algebraic surface and let

H = O(1) be a very ample line bundle on S. For a sheaf E on S, denote χ(S,E) =∑2i=0(−1)i dimH i(S,E) and let E(n) = E ⊗ H⊗n. Let r = rk(E) be the rank of

E. The discriminant of a coherent sheaf E is by definition [33, Section 3.4] the

characteristic class

∆(E) = c2 −r − 1

2rc2

1.

We do not distinguish between ∆(E) and its degree

d =

∫X

∆(E).

Definition 3.13. A torsion free sheaf E on S is H-semistable, if for all nonzero

subsheaves F ⊂ E, we have

χ(S, F (n))

rk(F (n))≤ χ(S,E(n))

rk(E(n)), for all n� 0.

It is called H-stable if the inequality is strict.

It can be shown that there exists a moduli space MS(r, c1, c2) of H-semistable

sheaves on S of rank r with Chern classes c1 and c2. The Hilbert scheme Hilbm(S)

of S is the special case with rank 1, c1 = 0 and c2 = m. For any torsion free and

H-semistable sheaf E Bogomolov’s inequality [33, Theorem 3.4.1] states that

d ≥ 0.

For H-stable sheaves this inequality is strict.

Let us restrict our attention to the case of rank 2. Then

d = c2 −c2

1

4,

and we denote the moduli spaces asMS(c1, d) =MS(2, c1, c2). The Euler character-

istics are again collected into a generating series

ZS,c1(q) =∑d

χ(MS(c1, d))qd.

Vafa and Witten interpret this generating function in [63] as the partition function

of a physical theory. The variable q is interpreted as q = e2πiτ , where τ is a coupling

parameter of the theory. To write it in this form, it has to be invariant under the

transformation T : τ → τ + 1. Furthermore, the theory should transform nicely

if one replaces the strong coupling with weak coupling. This corresponds to the

transformation S : τ → − 1τ. The operations S and T generate the action of SL(2,Z)

3.2. THE S-DUALITY CONJECTURE 21

on the upper half plane H. The S-duality conjecture desribes how the theory and

associated quantities transform with respect to the SL(2,Z)-action. In particular,

it predicts that, up to a fractional power of q, ZS,c1(q) has to be a meromorphic

modular form for a finite index subgroup of SL(2,Z).

It was observed by Balazs Szendroi that our formulae from Section 3.1 lead to

the following new modularity results, extending the results of [62] for type A.

Corollary 3.14. (S-duality for simple singularities) For type A and type D,

and, assuming Conjecture 3.7, for all types, the partition function ZC2/G∆(q) is, up

to a suitable fractional power of q, the q-expansion of a meromorphic modular form

of weight −12

for some congruence subgroup of SL(2,Z).

Proof. This follows straight from [62, Prop.3.2]. �

Corollary 3.15. (S-duality for surfaces with simple singularities) Let S be

a quasiprojective surface with simple singularities of type A and D, or, assuming

Conjecture 3.7, of arbitrary type. Then the generating function ZS(q) is, up to a

suitable fractional power of q, the q-expansion of a meromorphic modular form of

weight −χ(S)2

for some congruence subgroup of SL(2,Z).

Proof. Combine Theorem 3.12 and Corollary 3.14. �

CHAPTER 4

Abelian quotient singularities

This section gives all the proofs for the main results in the type A case. First, the

geometric problem is reduced to partition enumeration. Two independent solutions

to the enumerative problem are given. Finally a closely related case of cyclic quotient

singularities is treated.

4.1. Type A basics

Let ∆ be the root system of type An. Choosing a primitive (n + 1)-st root of

unity ω, the corresponding subgroup G∆ of SL(2,C), a cyclic subgroup of order

n+ 1, is generated by the matrix

σ =

(ω 0

0 ω−1

).

All irreducible representations of G∆ are one dimensional, and they are simply given

by ρj : σ 7→ ωj, for j ∈ {0, . . . , n}. The corresponding McKay quiver is the cyclic

Dynkin diagram of type A(1)n .

The group G∆ acts on C2; the quotient variety C2/G∆ has an An singularity at

the origin. The matrix σ clearly commutes with the diagonal two-torus T = (C∗)2,

and so T acts on the quotient C2/G∆ and the orbifold [C2/G∆]. Consequently T

also acts on the orbifold Hilbert scheme Hilb([C2/G∆]) and the (reduced) coarse

Hilbert scheme Hilb(C2/G∆) as well.

4.2. Partitions, torus-fixed points and decompositions

Consider the set N× N of pairs of non-negative integers; we will draw this set

as a set of blocks on the plane, occupying the non-negative quadrant. Label blocks

diagonally with (n+ 1) labels 0, . . . , n as in the picture; the block with coordinates

(i, j) is labelled with (i− j) mod (n+ 1). We will call this the pattern of type An,

or the diagonal labelling.

23

24 4. ABELIAN QUOTIENT SINGULARITIES

0 1 n−1 n

n 0 n−2n−1

0 1

n 0

1 2

0 1

......

. . . . . .

...

Let P denote the set of partitions. Given a partition λ = (λ1, . . . , λk) ∈ P , with

λ1 ≥ . . . ≥ λk positive integers, we consider its Young (or Ferrers) diagram, the

subset of N× N which consists of the λi lowest blocks in column i− 1. The blocks

in λ also get labelled (or colored) by the n+ 1 labels. Let Z∆ denote the resulting

set of diagonally labelled partitions, including the empty partition. For a labelled

partition λ ∈ Z∆, let wtj(λ) denote the number of blocks in λ labelled j, and define

the multiweight of λ to be wt(λ) = (wt0(λ), . . . ,wtn(λ)).

Proposition 4.1. The torus T acts with isolated fixed points on Hilb([C2/G∆]),

parametrized by the set Z∆ of diagonally labelled partitions. More precisely,

for k0, . . . , kn non-negative integers and ρ = ⊕nj=0ρ⊕kii , the T -fixed points on

Hilbρ([C2/G∆]) are parametrized by diagonally labelled partitions of multiweight

(k0, . . . , kn).

Proof. We just sketch the proof, which is well known [14, 18]. It is clear that

the T -fixed points on Hilb([C2/G∆]), which coincide with the T -fixed points on

Hilb(C2), are the monomial ideals in C[x, y] of finite colength. The monomial ideals

are enumerated in turn by Young diagrams of partitions. The labelling of each

block gives the weight of the G∆-action on the corresponding monomial, proving the

refined statement. �

Corollary 4.2. There exist a locally closed decomposition, depending on a

choice specified below, of Hilb([C2/G∆]) into strata indexed by the set of diagonally

labelled partitions. Each stratum is isomorphic to an affine space.

Proof. Again, this is well known [56]. Fixing a representation ρ, choose a

sufficiently general one-dimensional subtorus T0 ⊂ T which has positive weight on

both x and y. For general T0 ⊂ T , the fixed point set on Hilbρ([C2/G∆]) is unchanged

and in particular consists of a finite number of isolated points. Choosing positive

weights on x, y ensures that all limits of T0-orbits at t = 0 in Hilb([C2/G∆]) exist,

even though Hilbρ([C2/G∆]) is non-compact. Since Hilbρ([C2/G∆]) is smooth, the

result follows by taking the Bialynicki-Birula decomposition of Hilbρ([C2/G∆]) given

by the T0-action. �

4.2. PARTITIONS, TORUS-FIXED POINTS AND DECOMPOSITIONS 25

Denote by

Z∆(q0, . . . , qn) =∑λ∈Z∆

qwt(λ)

the generating series of diagonally labelled partitions, where we used multi-index

notation

qwt(λ) =n∏i=0

qwti(λ)i .

From either of the previous two statements, we immediately deduce the following.

Corollary 4.3. Let [C2/G∆] be a simple singularity orbifold of type A. Then

its orbifold generating series can be expressed as

(4.1) Z[C2/G∆](q0, . . . , qn) = Z∆(q0, . . . , qn).

Proposition 4.4. The generating series of diagonally labelled partitions has the

following form:

(4.2) Z∆(q0, . . . , qn) =

∑∞m=(m1,...,mn)∈Zk q

m11 · · · · · qmnn (q1/2)m

>·C·m∏∞m=1(1− qm)n+1

,

where q = q0 · · · · · qn and C is the (finite) Cartan matrix of type An. In particular,

(4.1) and (4.2) imply Theorem 3.1 for type A.

From the several existing proofs of this theorem we will give two below. The

first one, explained in 4.3, is an easy and direct combinatorial argument and was

found by the author [26]. The second one, which is summarized in 4.4, has originally

appeared in [18]. It has the advantage that is gives the starting point for the analysis

in type D.

We now turn to the coarse Hilbert scheme. Let us define a subset Z0∆ of the set of

diagonally labelled partitions Z∆ as follows. An diagonally labelled partition λ ∈ Z∆

will be called 0-generated (a slight misnomer, this should be really be “complement-

0-generated”) if the complement of λ inside N × N can be completely covered by

translates of N× N to blocks labelled 0 contained in this complement. Equivalently,

an diagonally labelled partition λ is 0-generated, if all its addable blocks (blocks

whose addition gives another partition) are labelled 0. It is immediately seen that

this condition is equivalent to the corresponding monomial ideal I � C[x, y] being

generated by its invariant part I ∩ C[x, y]G∆ . Indeed, we have the following.

Proposition 4.5. The torus T acts with isolated fixed points on Hilb(C2/G∆),

which are in bijection with the set Z0∆ of 0-generated diagonally labelled partitions.

More precisely, for a non-negative integer k, the T -fixed points on Hilbk(C2/G∆) are

parametrized by 0-generated diagonally labelled partitions λ with 0-weight wt0(λ) = k.

Proof. This is immediate from the above discussion. The T -fixed points of

Hilb(C2/G∆) are the monomial ideals I of C[x, y]G∆ of finite colength. Inside C[x, y],


the ideals they generate correspond to partitions which are 0-generated. The ring

C[x, y]G∆ has a basis consisting of monomials with corresponding blocks labelled 0

inside C[x, y]; thus the codimension of a monomial ideal I inside C[x, y]G∆ is simply

the number of blocks denoted 0. �

Denoting by

Z0∆(q) =

∑λ∈Z0

∆

qwt0(λ)

the corresponding specialization of the generating series of 0-generated diagonally

labelled partitions, we deduce the following.

Corollary 4.6. Let [C2/G∆] be a simple singularity orbifold of type A. Then

the coarse generating series can be expressed as

(4.3) ZC2/G∆(q) = Z0

∆(q).

Proof of Theorem 3.5 for the An case. The (dual) Coxeter number of

the type An root system is h∨ = n+ 1. Thus Theorem 3.5 for this case follows from

Corollary 4.6, formula (4.2), and the combinatorial Proposition 4.19 below, which

computes the series Z0∆(q). �

Finally, we investigate the higher rank equivariant case. There is a slight mod-

ification of the type An pattern where to each label, which are actually residue

classes modulo n + 1, a fixed modulo n + 1 residue class a is added. This again

induces a labelling on the Young diagrams which we will call this the diagonal

a-labelling. More generally, we can consider l-tuples of such Young diagrams λ. To

any vector a ∈ (Z/(n+ 1)Z)l of length l consisting of residue classes modulo n+ 1

we associate the diagonal ai-labelling on λi for each 1 ≤ i ≤ l. The is called the

diagonal a-labelling (or, shortly, a-labelling) of λ. As we will see immediately, the set

of such tuples appears in the analysis of the higher rank moduli spaces Mr([C2/G]).

Example 4.7. For n = 2 the diagonal (2, 1)-coloring on the Young diagrams

corresponding to the pair of partitions ((4, 3, 2), (2, 1, 1, 1)) is the following:2 0 1

1 2 0

0 1

2

, 1 2 0 1

0

.

We will denote the set of Young diagrams with the a-labelling as Z∆(a), and the set

of l-tuples of Young diagrams with the a-labelling as Z∆(a) for any a ∈ (Z/(n+ 1)Z)l.

For any 0 ≤ i ≤ n and λ ∈ Z∆(a) we denote by wti(a, λ), the i-weight of λ, which

is the number of boxes in λ, whose label according to the a-labelling is i. Clearly,

4.2. PARTITIONS, TORUS-FIXED POINTS AND DECOMPOSITIONS 27∑ni=0 wti(a, λ) = |λ| for any a ∈ (Z/(n+ 1)Z)l. We arrange the the i-weights into a

vector wt(a, λ) = (wt0(a), λ, . . . ,wtn(a, λ)) ∈ (Z≥0)n+1.

The colored (multivariable) generating series of a-labelled Young diagrams is

defined as

Z∆(a)(q) =∑

λ∈Z∆(a)

qwt(a,Y ) .

Similarly, the colored (multivariable) generating series of l-tuples of colored Young

diagrams is defined as

Z∆(a)(q) =∑

λ∈Z∆(a)

qwt(a,λ) .

Since tranforming the 0-labelling to the a-labelling corresponds to a change of

variables qi → qi+a, we obtain from Proposition 4.4

Corollary 4.8. The generating series of a-labelled partitions has the following

form:

(4.4) Z∆(a)(q0, . . . , qn) =

∑∞m=(m1,...,mn)∈Zk q

m11+a · · · · · qmnn+a(q

1/2)m>·C·m∏∞

m=1(1− qm)n+1,

where q = q0 · · · · · qn and C is the (finite) Cartan matrix of type An.

As a set Z∆(a) =∏l

m=1Z∆(am). Therefore, one immediately obtains

Corollary 4.9.

Z∆(a)(q) =l∏

m=1

Z∆(am)(q) .

In particular, the calculation of Z∆(a)(q) is easily reduced to that of Z∆(a)(q).

Let us fix a dimension vector w. It can be shown that there is a T = (C∗)|w|+2-

action on the associated quiver varieties M(v, w) for all v, whose fixed points are

isolated. Let a = (0, . . . , 0, . . . , n− 1, . . . , n− 1), where for each c ∈ C the number

of c’s in a is wc. Elements of a correspond in turn to basis vectors of W . The exact

order of the entries is not important for us since a permutation of them corresponds

to an automorphism of W on which, as mentioned above, the topology of M(v, w)

does not depend.

Proposition 4.10. [59, Proposition 5.7] The T -fixed points of M(v, w) are

indexed by |w|-tuples of diagonally colored Young diagrams λ such that |λ| = |v|, the

j-th diagram λj in λ is given the aj-coloring, and wt(a, λ) = v.

Corollary 4.11.

ZW∆ (q) =

∑v

χ(M(v, w))qv = Z∆(a)(q) .

Proof of Theorem 3.9. Putting together Corollary 4.8, Corollary 4.11 and

Corollary 4.9 gives the result. �


4.3. First approach: generalized Frobenius partitions

In this section we introduce a combinatorial method inspired by [12] generalizing

the ideas of Andrews [2] and we give a proof of Proposition 4.4 as developed in [26].

Definition 4.12. Two rows of nonnegative integers(f1 f2 . . . fdg1 g2 . . . gd

)are called a generalized Frobenius partition or F-partition of k if

k = d+d∑i=1

(fi + gi) .

Remark 4.13. A generalized Frobenius partition is a classical Frobenius partition

if moreover f1 > f2 > · · · > fd ≥ 0 and g1 > g2 > · · · > gd ≥ 0. In this case, we can

associate to the F-partition a Young diagram from which if we delete the d long

diagonal then the lengths of the rows below it are f1, f2, etc. and the length of the

columns above the diagonal are g1, g2, etc. This correspondence between Young

diagrams and classical F-partitions is bijective.

Let H be an arbitrary set consisting of finite sequences of nonnegative integer.

For arbitrary integers d and k let PH(k, d) denote the sequences in H of length d

which sum up to k. For any pair of such sets H1 and H2, let moreover PH1,H2(k) be

the number of generalized Frobenius partitions of k with elements in the first row

(f1, . . . , fr) from H1 and with elements in the second row (g1, . . . , gd) from H2. Then

the very useful result of Andrews says the following

Theorem 4.14 ([2], Section 3).∞∑k=0

PH1,H2(k)qk = [z0]∑k,m

PH1(k, d)qk(zq)d∑k,d

PH2(k, d)qkz−d ,

where [zm]∑Akz

k = Am.

The term qd in the first term of the right hand size corresponds to the contribution

of the diagonals. To have a more symmetric formula we will slightly change the

notions. Transform each generalized Frobenius partition(f1 f2 . . . fdg1 g2 . . . gd

)into (

f1 + 1 f2 + 1 . . . fd + 1

g1 g2 . . . gd

).

Then

k =d∑i=1

((fi + 1) + gi) .

4.3. FIRST APPROACH: GENERALIZED FROBENIUS PARTITIONS 29

For H an arbitrary set of sequences as above, let H ′ = {(f1 + 1, . . . , fd + 1) :

(f1, . . . , fd) ∈ H} be the shift of H by one upward. Then PH′(k, d) = PH(k − d, d),

and

(4.5)∞∑k=0

PH′1,H2(k)qk = [z0]

∑k,d

PH′1(k, d)qkzd∑k,d

PH2(k, d)qkz−d .

We aim for a multivariable generalization of Theorem 4.14. To be as general as

possible we introduce temporarily an arbitrary labelling (or coloring) set C. We will

specialize to the case of C = Z/(n+ 1)Z only later.

Let k be a vector of nonnegative integers indexed by the elements of C.

Definition 4.15. Two series of vectors consisting of integers and arranged into

two rows as (f

1f

2. . . f

d

g1

g2. . . g

d

)are called a colored generalized Frobenius partition or colored F-partition of k if

(1) the elements in fi

and gi

are indexed by the elements c ∈ C for each

1 ≤ i ≤ d;

(2) fi,c ≥ 0 and gi,c ≥ 0 for every 1 ≤ i ≤ d and c ∈ C;

(3)∑d

i=1(fi,c + gi,c) = kc for each c ∈ C.

We will call k =∑

c∈C kc =∑

c∈C∑d

i=1(fi,c + gi,c) the total weight of such a colored

F-partition.

At the moment we do not require any further relations between the elements

fi,c and gi,c but see end of this section and particularly Example 4.17 below, where

we apply this general construction to the enumeration of diagonally colored Young

diagrams.

Let H be an arbitrary set consisting of tuples of vectors, each of which is indexed

by elements of C. For an arbitrary vector k indexed by the elements of C and

consisting of nonnegative integers let PH(k, d) be the number of d-tuples of vectors

(f1, . . . , f

d) ∈ H which satisfy conditions (1) and (2) such that

∑di=1 fi,c = kc. If

both H1 and H2 are sets consisting of tuples of vectors, each element of which is

indexed by elements of C, then let PH1,H2(k) be the number of colored F-partition of

k in which the top row is in H1 and the bottom row is in H2.

Then the same ideas as that of Theorem 4.14 imply the following multivariable

analogue of (4.5).

Theorem 4.16.∑k

PH1,H2(k)qk = [z0]∑k,d

PH1(k, d)zdqk∑k,d

PH2(k, d)z−dqk .

We return finally to the setting of the orbifold Hilbert scheme, whose T-fixed

points are in bijection with diagonally labelled diagrams as explained in the beginning


of 4.2. We let C = Z/(n+ 1)Z be the set of labels which can appear in the pattern

of type An. To be able to apply Theorem 4.16 we first associate to each Young

diagram λ ∈ Z∆ a colored F-partition of k = {wtc(Y )}c∈C which uniquely describes

the diagonal coloring on λ. Assume that the main diagonal of Y consists of d blocks.

Then the associated colored F -partition is(f

1f

2. . . f

d

g1

g2. . . g

d

),

where fi,c is the number of blocks of color c in the i-th row below and including the

main diagonal, and gi,c is the number of blocks of color c in the i-th column above

the main diagonal for every 1 ≤ i ≤ d.

Example 4.17. The colored F-partition associated to the first diagram in Example

4.7 is ((1, 1, 1) (1, 0, 1)

(1, 1, 1) (0, 1, 0)

),

where each fi

= (fi,0, fi,1, fi,2) and each gi

= (gi,0, gi,1, gi,2).

For i = 1, 2 let Hi be the set of tuples of vectors which can appear as the i-th

row of a colored F-partition associated to a diagonally labelled Young diagram in

the above construction. Then

(4.6) Z∆,a(q) =∑k

PH1,H2(k)qk .

Lemma 4.18.

(1)

∑k

PH1(k, d)zdqk =∞∏k=0

n∏i=0

(1 + zq0 . . . qiqk) .

(2)

∑k

PH2(k, d)zdqk =∞∏k=0

n∏i=0

(1 + z−1qi+1 . . . qnqk) .

Proof. (1) It is clear that each term zq0 . . . qiqk corresponds to a part of a column

above and including the main diagonal which has length (n+ 1)k + i. Conversely,

the decomposition of each nonnegative number as (n+ 1)k + i is unique.

The proof of (2) is similar. �

4.3. FIRST APPROACH: GENERALIZED FROBENIUS PARTITIONS 31

The product of the two generating series in Lemma 4.18 is

(4.7)

∑k

PH1(k, d)zdqk ·∑k

PH2(k, d)zdqk

=∞∏k=0

n∏i=0

(1 + zq0 . . . qiqk)(1 + z−1qi+1 . . . qnq

k)

=∞∏k=1

n∏i=0

(1 + zq−1i+1 . . . q

−1n qk)(1 + (zq−1

i+1 . . . q−1n )−1qk−1)

=

(∞∏m=1

(1− qm)−1

)n n∏i=0

(∞∑

ji=−∞

(zq−1i+1 . . . q

−1n )jiq(

ji+12 )

),

where at the last equality we have used the following form of the Jacobi triple product

formula:

∞∏n=1

(1 + zqn)(1 + z−1qn−1) =

(∞∏n=1

(1− qn)−1

)∞∑

j=−∞

zjq(j+1

2 ) .

By (4.6) and Theorem 4.16 to obtain Z∆(q) we have to calculate the coefficient

of z0 in (4.7).

(4.8)

Z∆(q) = [z0]

(∞∏m=1

(1− qm)−1

)n+1 n∏i=0

(∞∑

ji=−∞

(zq−1i+1 . . . q

−1n )jiq(

ji+12 )

)

=

(∞∏m=1

(1− qm)−1

)n+1

·∑

j=(j0,...,jn)∈Zn+1∑i ji=0

q−j01 · · · · · q−j0−···−jn−1n q

∑ni=0 (ji+1

2 ) .

Let us introduce the following series of integers:

m1 = −j0 ,

m2 = −j0 − j1 ,...

mn = −j0 − j1 − · · · − jn−1 .

It is obvious that the map

{(j0, . . . , jn) ∈ Zn+1 :∑

i ji = 0} → Zn

(j0, . . . , jn) 7→ (m1, . . . ,mn)


is a bijection. The inverse of it is

j0 = −m1 ,

j1 = −m2 +m1 ,...

jn−1 = −mn +mn−1 ,

jn = mn .

If n = 1, then

(4.9)1∑i=0

(ji + 1

2

)=

(−m1 + 1

2

)+

(m1 + 1

2

)= m2

1 =1

2

(m> · C ·m

),

where C = (2) is the Cartan matrix of type A1.

If n > 1, then

(4.10)

n∑i=0

(ji + 1

2

)=

(−m1 + 1

2

)+

n−1∑i=1

(−mi+1 +mi + 1

2

)+

(mn + 1

2

)

= m21 + · · ·+m2

n −n−1∑i=1

mimi+1

=1

2

(m> · C ·m

),

where

C =

2 −1

−1 2 −1

−1 2. . .

−1

−1 2

is the Cartan matrix of type An.

Equations (4.8), (4.9) and (4.10) together immediately imply Proposition 4.4 for

all n > 0.

4.4. Second approach: abacus configurations

We now introduce a seemingly independent but also standard combinatorics

related to the type A root system, which will allow us to relate the generating

series Z∆ of diagonally labelled partitions to the specialized series Z0∆ of 0-generated

partitions. We follow the notations of [44].

The abacus of type An is the arrangement of the set of integers in (n+ 1) columns

according to the following pattern.

4.4. SECOND APPROACH: ABACUS CONFIGURATIONS 33

......

......

−2n− 1 −2n . . . −n− 2 −n− 1

−n −n+ 1 . . . −1 0

1 2 . . . n n+ 1

n+ 2 n+ 3 . . . 2n+ 1 2n+ 2...

......

...

Each integer in this pattern is called a position. For any integer 1 ≤ k ≤ n+ 1 the

set of positions in the k-th column of the abacus is called the k-th runner. An abacus

configuration is a set of beads, denoted by ©, placed on the positions, with each

position occupied by at most one bead.

To an diagonally labelled partition λ = (λ1, . . . , λk) ∈ Z∆ we associate its abacus

representation (sometimes also called Maya diagram) as follows: place a bead in

position λi − i+ 1 for all i, interpreting λi as 0 for i > k. Alternatively, the abacus

representation can be described by tracing the outer profile of the Young diagram of

a partition: the occupied positions occur where the profile moves “down”, whereas

the empty positions are where the profile moves “right”. In the abacus representation

of a partition, the number of occupied positive positions is always equal to the

number of absent nonpositive positions; we call such abacus configurations balanced.

Conversely, it is easy to see that any balanced configuration represents a unique

diagonally labelled partition, an element of Z∆.

For n = 0, we obtain a representation of partitions on a single runner; this is

sometimes called the Dirac sea representation of partitions.

The (n+ 1)-core of a labelled partition λ ∈ Z∆ is the partition obtained from λ

by successively removing border strips of length n+1, leaving a partition at each step,

until this is no longer possible. Here a border strip is a skew Young diagram which

does not contain 2× 2 blocks and which contains exactly one j-labelled block for all

labels j. The removal of a border strip corresponds in the abacus representation to

shifting one of the beads up on its runner, if there is an empty space on the runner

above it. In this way, the core of a partition corresponds to the bead configuration in

which all the beads are shifted up as much as possible; this in particular shows that

the (n+ 1)-core of a partition is well-defined. We denote by C∆ the set of (n+ 1)-core

partitions, and

c : Z∆ → C∆

the map which takes an diagonally labelled partition to its (n+ 1)-core.

Given an (n+ 1)-core λ, we can read the (n+ 1) runners of its abacus rrepresenta-

tion separately. These will not necessarily be balanced. The i-th one will be shifted

from the balanced position by a certain integer number ai steps, which is negative if

the shift is toward the negative positions (upwards), and positive otherwise. These

numbers satisfy∑n

i=0 ai = 0, since the original abacus configuration was balanced.


The set {a1, . . . , an} completely determines the partition, so we get a bijection

(4.11) C∆ ←→

{n∑i=0

ai = 0

}⊂ Zn+1.

We will represent an (n + 1)-core partition by the corresponding (n + 1)-tuple

a = (a0, . . . , an).

On the other hand, for an arbitrary partition, on each runner we have a partition

up to shift, so we get a bijection

Z∆ ←→ C∆ × Pn+1.

This corresponds to the structure of formula (4.2) above; its denominator is the

generating series of (n+ 1)-tuples of (unlabelled) partitions, whereas its numerator

(after eliminating a variable) is exactly a sum over a ∈ C∆. The multiweight of a

core partition corresponding to an element a is given by the quadratic expression

Q(a) in the exponent of the numerator of (4.2). For more details, see Bijections 1-2

in [19, §2].

Our purpose in the remaining part of this section is to prove the following,

completely combinatorial statement.

Proposition 4.19. Let ∆ be of type An, and let ξ be a primitive (n+ 2)-nd root

of unity. Then the generating series of 0-generated partitions can be computed from

that of all diagonally labelled ones by the following substitution:

Z0∆(q) = Z∆(q0, . . . , qn)

∣∣∣q0=ξ−nq,q1=···=qn=ξ

.

We start by combinatorially relating partitions to 0-generated partitions. Z0∆ is

clearly a subset of Z∆, but there is also a map

p : Z∆ → Z0∆

defined as follows: for an arbitrary partition λ, let p(λ) be the smallest 0-generated

partition containing it. Since the set of 0-generated partitions is closed under

intersection, p(λ) is well-defined, and it can be constructed as follows: p(λ) is the

complement of the unions of the translates of N × N to 0-labelled blocks in the

complement of λ. It is clear that p(λ) can equivalently be obtained by adding all

possible addable blocks to λ of labels different from 0.

Remark 4.20. The map p can also be described in the language of ideals. If the

monomial ideal I � C[x, y] corresponds to the partition λ, then the monomial ideal

i∗p∗I = (I ∩ C[x, y]G∆).C[x, y] � C[x, y] corresponds to the partition p(λ).

Lemma 4.21. The bead configurations corresponding to 0-generated partitions are

exactly those which have all beads right-justified on each row, with no empty position

4.4. SECOND APPROACH: ABACUS CONFIGURATIONS 35

to the right of a filled position. The map p : Z∆ → Z0∆ can be described in the abacus

representation by the process of pushing all beads of an abacus configuration as far

right as possible.

Proof. This follows from the description of the map from a partition to its

abacus representation using the profile of the partition. Indeed, a 0-generated

partition has a profile which only turns from “down” to “right” at 0-labelled blocks.

In other words, the only time when a string of filled positions can be followed by an

empty position is when the last filled position is on the rightmost runner. In other

words, there cannot be empty positions to the right of filled positions in a row. The

proof of the second statement is similar. �

Remark 4.22. As explained above, the maps c : Z∆ → C∆ and p : Z∆ → Z0∆

have natural descriptions on abacus configurations: c corresponds to pushing beads

all the way up within their column, whereas p corresponds to pushing beads all the

way to the right within their row. It is then clear that there is also a third map

Z∆ → 0Z∆ ⊂ Z∆, dual to p, defined on the abacus by pushing beads all the way to

the left. On labelled partitions this corresponds to the operation of removing all

possible blocks with labels different from 0. This dual constuction occured in the

literature earlier in [20].

Proof of Proposition 4.19. We will prove the substitution formula on the

fibres of the map p : Z∆ → Z0∆. In other words, we need to show that for any given

λ0 ∈ Z0∆, we have

(4.12)∑

µ∈p−1(λ0)

qwt(µ)∣∣∣q1=···=qn=ξ,q0=ξ−nq

= qwt0(λ0).

As a first step, we reduce the computation to 0-generated cores. Given an

arbitrary 0-generated partition λ, by the first part of Lemma 4.21 its core ν = c(λ)

is also 0-generated, and the corresponding abacus configuration can be obtained by

permuting the rows of the configuration of λ. Fix one such permutation σ of the

rows. Then, using the second part of Lemma 4.21, we can use the row permutation

σ to define a bijection

σ : p−1(λ)→ p−1(ν)

between (abacus representations of) partitions in the fibres, mapping λ itself to ν.

The difference between the partitions λ and ν is a certain number of border

strips, each removal represented by pushing up one bead on some runner by one step.

Each border strip contains one block of each label, so the total number of times we

need to push up a bead by one step on the different runners is N = wt0(λ)−wt0(ν).

Thus, with q = q0 · . . . · qn as in the substitution above, we can write

qwt(λ) = qwt0(λ)−wt0(ν)qwt(ν).


On the other hand, it is easy to see that in fact for any µ ∈ p−1(λ), the corresponding

σ(µ) can also be obtained by pushing up beads exactly N times, one step at a time,

the difference being just in the runners on which these shifts are performed. This

means that each µ differs from σ(µ) by the same number N = wt(λ) − wt(ν) of

border strips. Therefore, we have∑µ∈p−1(λ)

qwt(µ) = qwt0(λ)−wt0(ν)∑

µ∈p−1(ν)

qwt(µ).

This is clearly compatible with (4.12) and reduces the argument to 0-generated core

partitions.

Fix a 0-generated core λ ∈ Z0∆ ∩ C∆; using Lemma 4.21 again, the corresponding

(n+ 1)-tuple is a set of nondecreasing integers a = (a0, . . . , an) summing to 0. The

fibre p−1(λ) consists of partitions whose abacus representation contains the same

number of beads in each row as λ. The shift of one bead to the left results in the

removal in the partition of a block labelled i, with 1 ≤ i ≤ n. After substitution, this

multiplies the contribution of the diagram on the right hand side of (4.12) by ξ−1. If

we fix all but one row, which contains k beads, then these contributions add up to

n−k+1∑n1=0

n1∑n2=0

· · ·nk−1∑nk=0

(ξ−1)n1+···+nk =

(n+ 1

k

)ξ−1

,

where(mr

)z

= [m]z ![r]z ![m−r]z !

is the Gaussian binomial coefficient, with [m]z = 1−zm1−z .

The number of rows containing exactly k beads in the configuration corresponding

to λ is an+1−k − an−k. Therefore, the total contribution of the preimages, the left

hand side of (4.12), is

∑µ∈p−1(λ)

qwt(µ)∣∣∣q1=···=qn=ξ,q0=ξ−nq

=n∏k=1

(n+ 1

k

)an+1−k−an−k

ξ−1

qwt(λ)∣∣∣q1=···=qn=ξ,q0=ξ−nq

=n∏l=0

(( n+1n+1−l

)ξ−1(

n+1n−l

)ξ−1

)al

qwt(λ)∣∣∣q1=···=qn=ξ,q0=ξ−nq

=n∏l=0

(1− ξ−l−1

1− ξl−n−1

)alqwt(λ)

∣∣∣q1=···=qn=ξ,q0=ξ−nq

=n∏l=1

(1− ξ−n−1

1− ξ−1

1− ξ−l−1

1− ξl−n−1

)alqwt(λ)

∣∣∣q1=···=qn=ξ,q0=ξ−nq

= ξ−∑nl=1 lalqwt(λ)

∣∣∣q1=···=qn=ξ,q0=ξ−nq

,

4.5. OUTLOOK: CYCLIC QUOTIENT SINGULARITIES OF TYPE (P, 1) 37

where in the second equality we used(n+1

0

)z

=(n+1n+1

)z

= 1, in the penultimate equality

we used a0 = −a1 − · · · − an, and in the last equality we used

1− ξ−n−1

1− ξ−1

1− ξ−l−1

1− ξl−n−1= ξ−l,

which can be checked to hold for ξ a primitive (n+ 2)-nd root of unity. Incidentally,

as the multiplicative order of ξ is exactly n+ 2, all the denominators appearing above

are non-vanishing. Finally, according to [19, §2], we have

qwt(λ) = qQ(a)

2 qa1+···+an1 · . . . · qann ,

where again q = q0 · . . . · qn and Q : Zn → Z is the quadratic form associated to

C∆. Since q0 appears only in q on the right hand side, it is clear that Q(a)2

= wt0(λ).

Hence,

qQ(a)

2 qa1+···+an1 . . . qann

∣∣∣q1=···=qn=ξ

= qwt0(λ)ξ∑nl=1 lal .

This concludes the proof. �

4.5. Outlook: cyclic quotient singularities of type (p, 1)

This section is based on [27]. Recall the cyclic quotient singularities of type (p, 1)

from the end of 2.2. Let us fix the integer p once and for all. The surface singularity

X(p, 1) is again toric, i.e. it carries a (C∗)2-action with an isolated fixed point. On

the level of regular functions the fixed points are the monomials in

A := H0(OX(p,1)) = C[x, y]Zp ∼= C[xp, xp−1y, . . . , xyp−1, yp].

The (C∗)2-action lifts to Hilbn(X(p, 1)) for each n. The action of (C∗)2 on

Hilbn(X(p, 1)) again has only isolated fixed points which are given by the finite

colength monomial ideals in A.

As mentioned before, finite colength monomial ideals inside C[x, y] are in one-to-

one correspondence with partitions and with Young diagrams. The generators of a

monomial ideal are the functions corresponding to those blocks in the complement

of the diagram which are at the corners. Since A ⊂ C[x, y], each fixed point of the

(C∗)2-action on Hilbn(X(p, 1)) corresponds to again a Young diagram.

A block at position (i, j) is called a 0-block (for the singularity X(p,1)) if i+ j ≡0 (mod n). We will call a Young diagram 0-generated (for the singularity X(p,1)) if

its generator blocks are 0-blocks. The 0-weight of a (not necessarily 0-generated)

Young diagram λ is the number of 0-blocks inside the Young diagram. It is denoted

as wt0(λ). Let P0 be the set of 0-generated Young diagrams. It decomposes as

P0 =⊔n≥0

P0(n),

where P0(n) is the set of 0-generated Young diagrams which have 0-weight n.


Example 4.23. If p = 3, then the following is a 0-generated Young diagram (we

also indicated the generating 0-blocks):

0 0

0

0

0

0

0

The 0-weight of this Young diagram is 3.

The generators of any ideal of A, when considered as functions in C[x, y], have

to be invariant under the Zp-action. This implies the following statement.

Lemma 4.24. The monomial ideals inside A of colength n are in one-to-one

correspondence with the 0-generated Young diagrams in P0(n).

Corollary 4.25.

ZX(p,1)(q) =∑λ∈P0

qwt0(λ) =∑n≥0

|P0(n)|qn,

where |S| denotes the number of elements in the set S.

For any 0-generated Young diagram, we can consider the area which is between

the diagram and the line x+ y = c. Here c is an integer congruent to 0 modulo p and

it is as small as possible such that the line x+ y = c does not intersect the diagram.

In other words, this line is just the antidiagonal closest to the diagram. The line cuts

out the smallest isosceles right-angled triangle which contains the whole diagram. In

the case of Example 4.23, this looks as follows:

0 0

0

0

0

0

0

0

0

0

0

0

If, for example, p = 1 then the area between the diagram, the x and y coordinate

axes, and the above mentioned antidiagonal, when rotated 45 degrees counterclockwise

and flipped, is a special type of a fountain of coins as introduced in [58]. An (n, k)

fountain (of coins) is an arrangement of n coins in rows such that there are exactly k

consecutive coins in the bottom row, and such that each coin in a higher row touches

exactly two coins in the next lower row. In our case, the 0-blocks in the area under

consideration (which is colored grey in the diagram above) are replaced by coins or

circles or zero symbols.

We generalize this notion to arbitrary p. An (n, k) p-fountain is an arrangement

of n coins in rows such that


• there are exactly k consecutive coins in the bottom row,

• immediately below each coin there are exactly p+1 “descendant” coins in

the next lower row,

• and p coins among the descendants of two neighboring coins coincide.

In other words, we rotate and flip the area between the axes and a specific antidiagonal,

and the coins can be placed exactly on the 0-blocks, such that if there is a coin

somewhere, then there has to be coins on all the 0-blocks in the area which have

higher x or y coordinates in the original orientation of the plane. The empty diagram

is considered as a (0, 0) p-fountain.

Continuing the case of Example 4.23 further, the associated (9, 7) 3-fountain is

00

0000000

Here we also indicated the descendants of the coins in the upper row.

An (n, k) p-fountain is called primitive if its next-to-bottom row contains no

empty positions, i.e. contains k − p coins. In particular, the fountain with p coins in

the bottom row but with no coin in the higher rows is primitive, but the ones with

less than p coins in the bottom row are not primitive. The fountains that appear

between our 0-generated Young diagrams and the diagonals are special because in

each case there is at least one empty position in the next-to-bottom row, so they

correspond exactly to the non-primitive p-fountains.

Let f(n, k) (reps., g(n, k)) be the number of arbitrary (resp., primitive )(n, k)

p-fountains. Let F (q, z) =∑

n,k≥0 f(n, k)qnzk (resp., G(q, z) =∑

n,k≥0 g(n, k)qnzk)

be the two variable generating function of the sequence f(n, k) (resp., g(n, k)). We

will calculate F (q, z) and G(q, z) by extending the ideas of [58].

By removing the bottom row of a primitive (n, k) p-fountain one obtains a

(n− k, k − p) p-fountain. Therefore,

g(n, k) = f(n− k, k − p) (n ≥ k, k ≥ p),

and

(4.13) G(q, z) = (qz)pF (q, qz).

We prescribe that

(4.14) f(0, 0) = · · · = f(p− 1, p− 1) = 1 and g(0, 0) = · · · = g(p− 1, p− 1) = 0.

Let us consider an arbitrary (n, k) p-fountain F , and assume that the first empty

position in the next-to-bottom row is the r-th (1 ≤ r ≤ k− p+ 1). Then we can split

F into a primitive (m+ p− 1, r + p− 1) p-fountain and a not necessarily primitive

(n−m, k − r) p-fountain after the first and before the last descendant coin of the


above mentioned missing r-th position. The descendant coins at the second to the

penultimate positions will be doubled.

Sticking to our favorite Example 4.23, the splitting looks as follows:

00

00000000 ( )⇒

00

00000+

0000

The dashed line on the left picture indicates the missing coin from the second row

and its descendants. The primitive part is to the left of the right parenthesis, while

the remaining part is to the right of the left parenthesis.

This factorization is unique.

(4.15) f(n, k) =∑

0≤m≤n−p+10≤r≤k−p+1

g(m+ p− 1, r + p− 1)f(n−m, k − r) (n, k ≥ p).

The conditions m ≤ n− p + 1 and r ≤ k − p + 1 are equivalent to p− 1 ≤ n−mand p− 1 ≤ k − r respectively. That is, the other remaining fountain has to have a

first row of length at least p− 1. From (4.14) we see that F (q, z) has the form

F (q, z) = 1 + qz + · · ·+ (qz)p−1 + . . . .

Then F (q, z)− 1− qz − · · · − (qz)p−2 is exactly the generating function of fountains,

which have a first row of length at least p−1, that is, which can appear as the second

factor on the right hand side of (4.15). Therefore,

(qz)−p+1G(q, z)(F (q, z)− 1− qz − · · · − (qz)p−2)

enumerates all p-fountains for which n, k ≥ p. The generating function of all

p-fountains then satisfies

(4.16)

F (q, z) = 1 + qz + · · ·+ (qz)p−1 + (qz)−p+1G(q, z)(F (q, z)− 1− qz − · · · − (qz)p−2).

Using (4.16),

(F (q, z)− 1− qz − · · · − (qz)p−2)(1− (qz)−p+1G(q, z)) = (qz)p−1.

Then, by (4.13), the generating function of p-fountains is

(4.17)

F (q, z) =(qz)p−1

1− (qz)−p+1G(q, z)+ 1 + qz + · · ·+ (qz)p−2

=(qz)p−1

1− qzF (q, qz)+

1− (qz)p−1

1− qz

=(qz)p−1

1− qz(

(q2z)p−1

1−q2zF (q,q2z)+ 1−(q2z)p−1

1−q2z

) +1− (qz)p−1

1− qz= . . .

=(qz)p−1

1− qz(

(q2z)p−1

1−q2z(

(q3z)p−1

1−... +1−(q3z)p−1

1−q3z

) + 1−(q2z)p−1

1−q2z

) +1− (qz)p−1

1− qz.


Consequently, the generating function of primitive p-fountains is

(4.18) G(q, z) =(q2z)p−1

1− q2z

((q3z)p−1

1−q3z(

(q4z)p−1

1−... +1−(q4z)p−1

1−q4z

) + 1−(q3z)p−1

1−q3z

) +1− (q2z)p−1

1− q2z.

Remark 4.26. For p = 1, Ramanujan [3, p. 104] obtained the beautiful formula

F (q, z) =1

1− qz

1− q2z1−...

=

∑n≥0(−qz)n qn

2

(1−q)(1−q2)...(1−qn)∑n≥0(−z)n qn2

(1−q)(1−q2)...(1−qn)

.

The number of non-primitive (n, k) p-fountains is obviously

h(n, k) = f(n, k)− g(n, k),

which gives

(4.19) H(q, z) = F (q, z)−G(q, z) = F (q, z)− (qz)pF (q, qz)

for the generating series H(q, z) of the numbers h(n, k).

Proof of Theorem 3.10. As mentioned above, we augment each 0-generated

Young diagram to the smallest isosceles right-angled triangle. The area between

the diagram and the triangle will be a p-fountain. For a fixed p, the hypotenuse

of the possible isosceles right-angled triangles contains lp + 1 blocks, where l is a

non-negative integer. The number of 0-blocks in the triangle with lp + 1 blocks

on the hypotenuse is∑

0≤i≤l ip + 1 = p l(l+1)2

+ l + 1. Therefore, the two variable

generating series of these triangles is∑l≥0

qpl(l+1)

2+l+1zlp+1.

We will see immediately that adding the terms with negative l will not affect the

final result. Hence, we define

(4.20)

T (q, z) =∞∑

l=−∞

qpl(l+1)

2+l+1zlp+1 = (qz)

∞∑l=−∞

(qp)(l+12 )(qzp)l

= (qz)∞∏n=1

(1 + zpqnp+1)(1 + z−pq(n−1)p−1)(1− qnp),

where at the last equality we have used the following form of the Jacobi triple product

identity:∞∏n=1

(1 + zqn)(1 + z−1qn−1)(1− qn) =∞∑

j=−∞

zjq(j+1

2 ).

The generating series of 0-generated Young diagrams is then

(4.21)∑λ∈P0

qwt0(λ) = [z0]T (q, z)H(q−1, z−1).


Taking the coefficient of z0 ensures that the hypotenuse of the triangle and the

bottom row of the p-fountain match together, and also that the terms of T (q, z) with

negative powers of z do not contribute into the result. Putting together Corollary

4.25, (4.19)and (4.21) concludes the proof. �

CHAPTER 5

Type Dn: ideals and Young walls

This chapter starts to develop the tools for proofs in the type D case. Young

walls, which are the analogs of diagonally labelled Young diagrams of the type A

case, are introduced. Their building blocks are shown to be in bijection with cells

of the equivariant Grassmannian. Finally, with each invariant ideal a Young wall is

associated.

5.1. The binary dihedral group

Fix an integer n ≥ 4, and let ∆ be the root system of type Dn. For ε a fixed

primitive (2n− 4)-th root of unity, the corresponding subgroup G∆ of SL(2,C) can

be generated by the following two elements σ and τ :

σ =

(ε 0

0 ε−1

), τ =

(0 1

−1 0

).

The group G∆ has order 4n− 8, and is often called the binary dihedral group. We

label its irreducible representations as shown in Table 1. There is a distinguished

2-dimensional representation, the defining representation ρnat = ρ2. See [35, 10] for

more detailed information.

We will often meet the involution on the set of representations of G∆ which

is given by tensor product with the sign representation ρ1: on the set of indices

{0, . . . , n}, this is the involution j 7→ κ(j) which swaps 0 and 1 and n − 1 and n,

fixing other values {2, . . . , n− 2}. Given j ∈ {0, . . . , n}, we denote κ(j, k) = κkn(j);

this is an involution which is nontrivial when k and n are odd, and trivial otherwise.

The special case k = 1 will also be denoted as j = κn(j).

The following identities will be useful:

(5.1) ρ⊗2n−1∼= ρ⊗2

n∼= ρ0, ρn−1⊗ρn ∼= ρ1, ρ1⊗ρn−1

∼= ρn, ρ1⊗ρn ∼= ρn−1, ρ⊗21∼= ρ0.

5.2. Young wall pattern and Young walls

We describe here the type D analogue of the set of labelled partitions used in

type A, following [39, 41]. In this section, we only describe the combinatorics; see

2.4 for the representation-theoretic significance of this set.

43

44 5. TYPE DN : IDEALS AND YOUNG WALLS

ρ Tr(1) Tr(σ) Tr(τ)

ρ0 1 1 1

ρ1 1 1 −1

ρ2 2 ε+ ε−1 0...

...

ρn−2 2 εn−3 + ε−(n−3) 0

ρn−1 1 −1 −in

ρn 1 −1 in

Table 1. Labelling the representations of the group G∆

First we define the Young wall pattern of type1 Dn, the analogue of the (n+ 1)-

labelled positive quadrant lattice of type An used above. This is the following

infinite pattern, consisting of two types of blocks: half-blocks carrying possible labels

j ∈ {0, 1, n− 1, n}, and full blocks carrying possible labels 1 < j < n− 1:

2

n−2

n−2

2

2

...

...

2

n−2

n−2

2

2

...

...

2

n−2

n−2

2

2

...

...

2

n−2

n−2

2

2

...

...

2

n−2

n−2

2

2

...

...

2

n−2

n−2

2

2

...

...

2

n−2

n−2

2

2

...

...

2

n−2

n−2

2

2

...

...

01

n−1n

01

01

n−1n

01

01

n−1n

01

01

n−1n

01

10

nn−1

10

10

nn−1

10

10

nn−1

10

10

nn−1

10

. . .

...

Next, we define the set of Young walls2 of type Dn. A Young wall of type Dn is a

subset Y of the infinite Young wall of type Dn, satisfying the following rules.

(YW1) Y contains all grey half-blocks, and a finite number of the white blocks and

half-blocks.

(YW2) Y consists of continuous columns of blocks, with no block placed on top of

a missing block or half-block.

1The combinatorics of this section should really be called type D(1)n , but we do not wish to

overburden the notation. Also we have reflected the pattern in a vertical axis compared to the

pictures of [39, 41].2In [39, 41], these arrangements are called proper Young walls. Since we will not meet any

other Young wall, we will drop the adjective proper for brevity.

5.3. DECOMPOSITION OF C[X,Y ] AND THE TRANSFORMED YOUNG WALL PATTERN 45

(YW3) Except for the leftmost column, there are no free positions to the left of

any block or half-block. Here the rows of half-blocks are thought of as two

parallel rows; only half-blocks of the same orientation have to be present.

(YW4) A full column is a column with a full block or both half-blocks present at

its top; then no two full columns have the same height3.

Let Z∆ denote the set of all Young walls of type Dn. For any Y ∈ Z∆ and label j ∈{0, . . . , n} let wtj(Y ) be the number of white half-blocks, respectively blocks, of label

j. These are collected into the multi-weight vector wt(Y ) = (wt0(Y ), . . . , wtn(Y )).

The total weight of Y is the sum

|Y | =n∑j=0

wtj(Y ),

and for the formal variables q0, . . . , qn,

qwt(Y ) =n∏j=0

qwtj(Y )j .

5.3. Decomposition of C[x, y] and the transformed Young wall pattern

The group G∆ acts on the affine plane C2 via the defining representation ρnat = ρ2.

Let S = C[x, y] be the coordinate ring of the plane, then S = ⊕m≥0Sm where Sm is

the mth symmetric power of ρnat, the space of homogeneous polynomials of degree

m of the coordinates x, y.

We further decompose

Sm =n⊕j=0

Sm[ρj]

into subrepresentations indexed by irreducible representations. We will also use this

notation for linear subspaces: for U ⊂ Sm a linear subspace, U [ρj ] = U ∩ Sm[ρj ]. We

will call an element f ∈ S degree homogeneous, if f ∈ Sm for some m; we call it

degree and weight homogeneous, if f ∈ Sm[ρj] for some m, j.

The decomposition of S into G∆-summands can be read off very conveniently from

the transformed Young wall pattern. The transformation is an affine one, involving a

shear: reflect the original Young wall pattern in the line x = y in the plane, translate

the nth row by n to the right, and remove the grey triangles of the original pattern.

In this way, we get the following picture:

3This is the properness condition of [39].


...

. . .

2 n−2 n−2 2 2. . . . . .

2 n−2 n−2 2 2. . . . . .

2 n−2 n−2 2 2. . . . . .

2 n−2 n−2 2 2. . . . . .

0 n−1n 0

1

0 n−1n 0

1

1 nn−1 1

0

1 nn−1 1

0

As it can be checked readily, this is a representation of S and its decomposition into

G∆-representations. The homogeneous components Sm are along the antidiagonals.

For 1 < i < n− 1, a full block labelled j below the diagonal, together with its mirror

image, correspond to a 2-dimensional representation ρj. For j ∈ {0, 1, n− 1, n}, a

full block labelled j on the diagonal, as well as a half-block labelled j below the

diagonal with its mirror image, corresponds to a one-dimensional representation.

The dimension of Sm[ρj] is the same as the total number of full blocks labelled j on

the mth diagonal in the transformed Young wall pattern, counting mirror images

also.

It is easy to translate the conditions (YW1)-(YW4) into the combinatorics of

the transformed pattern; see Proposition 5.7 and Remark 5.8 below. Pictures of

some small Young walls in the transformed pattern can be found in Examples 6.5-6.9

below.

5.4. Subspaces and operators

For each non-negative integer m and irreducible representation ρj, consider the

space Pm,j of nontrivial G∆-invariant subspaces of minimal dimension in Sm[ρj].

Specifically, if ρj is one-dimensional, then these will be lines, and Pm,j is simply the

projectivization PSm[ρj]. If ρj is two-dimensional, then Pm,j is a closed subvariety

of Gr(2, Sm[ρj]). It is easy to see that in this case also, Pm,j is isomorphic to a

projective space.

More generally, let Grm,j be the space of (r − 1)-dimensional projective subspaces

of Pm,j . If ρj is one-dimensional, then this is the Grassmannian Gr(r, Sm[ρj ]). When

ρj is two-dimensional, then Grm,j is a closed subvariety of Gr(2r, Sm[ρj]) isomorphic

to a Grassmannian of rank r. Clearly G1m,j = Pm,j.

For 0 ≤ j ≤ n, we introduce operators Lj : Gr(S)→ Gr(S) on the Grassmannian

Gr(S) of all linear subspaces of the vector space S as follows: for v ∈ Gr(S), we set

(1) L0v = v;

(2) L1v = xy · v;

(3) for 1 < j < n− 1, Ljv = 〈xj−1 · v, yj−1 · v〉;(4) Ln−1v = (xn−2 − inyn−2) · v;

(5) Lnv = (xn−2 + inyn−2) · v.

5.5. CELL DECOMPOSITIONS OF EQUIVARIANT GRASSMANNIANS 47

Sometimes we will use the notation L2 = L2,x + L2,y for the x- and y-component

of the operator L2, i.e. multiplication with x, respectively y. The operators

above restrict to operators L0 : Gr(Sm) → Gr(Sm), L1 : Gr(Sm) → Gr(Sm+2),

Lj : Gr(Sm)→ Gr(Sm+j−1) for 1 < j < n− 1, and Ln−1, Ln : Gr(Sm)→ Gr(Sm+n−2)

on the Grassmannians of the graded pieces Sm. To simplify notation, if we do not

write the space to which these operators are applied, then application to 〈1〉 is meant.

So, for example, the symbol L21 standing alone denotes the vector subspace 〈x2y2〉

of S4, while L2 alone denotes the two-dimensional vector subspace 〈x, y〉 of S1. For

a linear subspace v of S, the sum∑

j∈I Ljv denotes the subspace of S generated

by the images Ljv. We use the operator notation also for a set of subspaces; the

meaning should be clear from the context.

5.5. Cell decompositions of equivariant Grassmannians

We start this section by defining decompositions of the Grassmannians Pm,j of

nontrivial G∆-invariant subspaces of minimal dimension in Sm[ρj]. Given (m, j), let

Bm,j denote the set of pairs of non-negative integers (k, l) such that k+ l = m, l ≥ k,

and the block position (k, l) on the m-th antidiagonal on or below the main diagonal

contains a block or half-block of color j. Here k is the row index, l is the column

index, and both of them in a nonnegative integer. It clearly follows from this setup

that

dimPm,j = |Bm,j| − 1.

Proposition 5.1. Given (m, j), there exists a locally closed stratification

Pm,j =⊔

(k,l)∈Bm,j

Vk,l,j,

which is a standard stratification of the projective space Pm,j into affine spaces Vk,l,jof decreasing dimension.

We will call Vk,l,j the cells of Pm,j . The decomposition will be defined inductively,

based on the following Lemma. Recall that j 7→ κ(j) denotes the involution on

{0, . . . , n} which swaps 0 and 1 and n− 1 and n.

Lemma 5.2. For any l ≥ 0 and any j ∈ [0, n], we have an injection

L1 : Pl−2,j → Pl,κ(j).

This map is an isomorphism except in the case when the block or half-block in the

bottom row of the transformed Young wall pattern on the l-th antidiagonal has label

j, in which case the image has codimension one.

Proof. It is clear that multiplication by L1 induces an injection, so we simply

need to check the dimensions. The statement then clearly follows by looking at the

transformed Young wall pattern: multiplication by L1 corresponds to shifting the


(l−2)-nd diagonal up by one diagonal step to the l-th diagonal; the number of blocks

or half-blocks labelled j is identical, unless the new (half-)block has label j, and then

the codimension is exactly one. �

Remark 5.3. The half-block in the bottom row of the transformed Young

wall pattern in the l-th antidiagonal has label j = 0, 1 for l ≡ 0 mod (2n − 4)

except at (0, 0) where only 0 occurs. Half-blocks labelled j = (n − 1), n occur for

l ≡ n − 2 mod (2n − 4). For j ∈ [2, n − 2], there are full blocks labelled j in the

bottom row on antidiagonals for l ≡ j − 1 or 2n− 3− j mod (2n− 4).

Proof of Proposition 5.1. Nontrivial cells V0,l,j need to be defined exactly

when the block or half-block in the bottom row of the transformed Young wall

pattern in the l-th antidiagonal has label j. In these cases, we set the cells along the

bottom row to be

V0,l,j = Pl,j \ L1Pl−2,κ(j).

Once the cells V0,k,j along the bottom row are defined, we define the general cells for

all 0 ≤ j ≤ n, all l and k by

Vk,k+l,j = Lk1V0,l,κk(j).

What this says is that the cells are shifted up diagonally by L1, taking into account

that L1 multiplies by the sign representation, so shifts the indices by the appropriate

power of the involution κ. By induction, we obtain a decomposition of Pm,j with the

stated properties. �

As it is well known, a decomposition of a projectivization of a vector space into

affine cells is equivalent to giving a flag in the space itself. This induces a natural

decomposition of all higher rank Grassmannians into Schubert cells, which are known

to be affine. Thus our cell decomposition of Pm,j induces cell decompositions of all

Grm,j. Since the cells in the first decomposition are indexed by the set Bm,j, the

cells in the second will be indexed by subsets of Bm,j of size r. A Schubert cell of

Grm,j corresponding to a subset S = {(k1, l1), . . . (kr, lr)} ⊂ Bm,j will consist of those

(r − 1)-dimensional projective subspaces of Pm,j which intersect Vki,li,j nontrivially

for all 1 ≤ i ≤ r. We will denote the cell corresponding to S in Grm,j by VS,j. We

obtain a locally closed decomposition

Grm,j =

⊔S⊆Bm,j|S|=r

VS,j.

Occasionally, when it is clear from the context that S is a subset of Bm,j, we will

supress the index j and write just VS for the Schubert cells of Grm,j.

We will call a Schubert cell maximal if it intersects the maximal dimensional

cell of Pm,j nontrivially. Such a cell corresponds to subsets S ⊂ Bm,j which contain

(kmin, l) where kmin is minimal among the first components of the elements of Bm,j.

5.5. CELL DECOMPOSITIONS OF EQUIVARIANT GRASSMANNIANS 49

The intersection with Vkmin,l,j of a subspace corresponding to a point in a maximal

Schubert cell is an affine subspace of Vkmin,l,j. Conversely, to any affine subspace of

Vkmin,l,j , there corresponds a point in a maximal Schubert cell given by the completion

of the subspace in Pm,j.

For a maximal subset S, denote by S ⊂ Bm,j the set of indices which we get

by deleting (kmin, l) from S. S is empty, if |S| = 1. Define the codimension one

projective subspace

Pm,j =⊔

{(k,l)∈Bm,j : k>kmin}

Vk,l,j ⊂ Pm,j = Pm,j \ Vkmin,l,j.

For each (r− 1)-dimensional subspace U ⊂ Pm,j intersecting the affine space Vkmin,l,jnontrivially, let U = U ∩ Pm,j.

Lemma 5.4. The map ω : VS,j → VS,j defined by ω(U) = U is a trivial affine

fibration with fibre A|Bm,j |−|S|.

We can think of this map as associating to an affine subspace of Vkmin,l,j its set

of “ideal points at infinity”.

Consider the fibre ω−1(U) over a point U ∈ VS, which we will also denote by

VS|U below. This fibre consists of those subspaces U ⊆ Pm,j which intersect Pm,j in

U , i.e. when considered as an affine subspace of Vkmin,l,j, they have U as their set of

“points at infinity”. We will denote the set of such subspaces also by Vkmin,l,j/U . This

notation means that we take the cosets in Vkmin,l,j of an arbitrary affine subspace

U ⊂ Vkmin,l,j with U = Pm,j∩U . The affine structure on Vkmin,l,j descends to an affine

structure on Vkmin,l,j/U which does not depend on the particular affine subspace U

whose cosets were taken.

We define the mapping ω : VS,j → VS,j as the identity for those index sets and

the corresponding cells which are not maximal. In such cases, S = S considered as a

subset of Bm,j \ {(kmin, l)}.We also need a description of the affine subspaces of the cells Vk,l,j for k > kmin.

The relevant Schubert cells in this case are indexed by those subsets S of Bm,j which

contain (k, l) but do not contain any (k′, l′) for k′ < k. Hence, the index set Bm,j is

first truncated by deleting the pairs (k′, l′) with k′ < k. We will denote the result

as Bm,j(k). Then, the maximal Schubert cells for Vk,l,j correspond to those subsets

S ⊆ Bm,j(k) of the truncated index set which contain (k, l). For these, S is defined

by removing (k, l) from S. There is still a morphism ω : VS,j → VS,j which is defined

in the same way as above; its global structure and the description of its fibres is

analogous to the previous special case.

Note that the notation S is ambiguous at this point: any maximal subset

S ⊆ Bm,j(k) can also be considered as a nonmaximal subset of Bm,j(k′) for k′ < k.

If we view S as a subset of Bm,j(k), then S = S \ {(k,m− k)}. On the other hand,

if we view it as a nonmaximal subset of Bm,j(k′) for k′ < k, then S = S. We have


decided not to introduce extra notation; when this notation gets used below, we will

always specify the reference point k explicitly.

5.6. The Young wall associated with a homogeneous ideal

In this section, we study ideals generated by degree- and weight-homogeneous

polynomials; we will call such ideals simply homogeneous ideals. Here is the main

definition of this section.

Definition 5.5. Consider a homogeneous G∆-invariant ideal I � C[x, y]. Let

YI denote the following subset of the transformed Young wall pattern of type Dn:

for each block or half-block (k, l) of label j, with k + l = m, include this block or

half-block in YI if and only if I ∩ Sm[ρj ] does not intersect the preimage in Sm[ρj ] of

the stratum Vk,l,j ⊂ Pm,j from the stratification of Proposition 5.1. YI will be called

the profile of I.

It will be useful to introduce a little bit of extra notation, and to reformulate this

definition in the new notation. Given a homogeneous G∆-invariant ideal I, let Im,jbe the set of G∆-invariant subspaces of minimal dimension in I ∩ Sm[ρj]. Then as

I ∩ Sm[ρj] ⊂ Sm[ρj] is a linear subspace, Im,j ⊂ Pm,j is a projective linear subspace.

Then the definition simply says that a block or half-block (k, l) labelled j is included

in YI if and only if Im,j ∩ Vk,l,j = ∅, for m = k + l as before. Since {Vk,l,j} is a

standard stratification of the projective space Pm,j into affine spaces, {Im,j ∩Vk,l,j} is

also a standard stratification of its projective linear subspace Im,j into affine spaces,

and so has the same number of strata as its affine dimension. We conclude

Lemma 5.6. For all m, j, the number of absent blocks or half-blocks of label j on

the m-th diagonal equals dim(I ∩ Sm[ρj]).

Proposition 5.7. Given a homogeneous G∆-invariant ideal I � C[x, y], the

associated subset YI of the transformed Young wall pattern of type Dn has the

following properties.

(1) If a full or half block is missing, then all the blocks above-right from it on

the diagonal are missing.4

(2) If a full block is missing, then all full or half blocks to the right of it are

missing, and at least one (full or half) block immediately above it is missing.

(3) If a half block is missing, then the full block to the right of it is missing.

(4) If both half-blocks sharing the same block position are missing, then the full

block immediately above this position is missing.

In particular, if I is of finite codimension, then YI is a Young wall of type Dn, an

element of the set Z∆.

4Again, for a missing half block only the half blocks of the same orientation have to be missing.

5.6. THE YOUNG WALL ASSOCIATED WITH A HOMOGENEOUS IDEAL 51

Remark 5.8. As it can be checked from the definitions, the relationship between

the directions in the original and transformed Young wall patterns is the following:

(right, up, diagonally right and down) in the original correspond after transformation

to (diagonally right and up, right, up) respectively. This way, it is easy to check the

correspondence between the rules for Young walls from 5.2 and this proposition.

Proof of Proposition 5.7. Fix a homogeneous invariant ideal I � C[x, y]

and let YI be the corresponding subset of the Young wall pattern. Property (1) of

YI follows by applying L1, recalling the inductive nature of the stratification of Pm,jusing L1. The inductice construction also implies that it suffices to check properties

(2)-(4) for blocks missing on the bottom row.

Let us next prove (2) in the general case, when a full block in position (0, l) in

representation j ∈ [3, n− 3] is missing from YI ; by the choice of j, both above and to

the right of this block there are also full blocks. Since the block at (0, l) is missing,

there is an invariant 2-dimensional subspace u ∈ Il,j ∩ V0,l,j contained in I. Since u

is in the lowest stratum V0,l,j of Pl,j, it has a basis one of whose members at least is

not divisible by xy; without loss of generality, we may assume that this polynomial

is xap where a is a non-negative integer and p a polynomial in x, y not divisible by

x, y. Now we can write

L2u = u+ ⊕ u−

with u+ ∈ Il+1,j+1 and u− ∈ Il+1,j−1. Then u+ must contain a polynomial with xa+1p

as nonzero summand, so it cannot be in the image of L1; so we have u+ ∈ V0,l+1,j+1.

Similarly, u− must contain a polynomial with xayp as nonzero summand, so it cannot

be in the image of L21 and so u− ∈ V1,l,j−1. Thus indeed both the blocks in positions

(0, l + 1) and (1, l) are missing as claimed.

Let us now consider what changes if j is chosen such that there are half-blocks

around. Suppose first that the half-blocks happen to lie to the right of our block

labelled j. Then we have a decomposition

L2u = u1+ ⊕ u2

+ ⊕ u−,

with ui+ both one-dimensional. In this case, it is easy to check that the polynomial

xa+1p cannot itself generate a one-dimensional eigenspace, so both ui+ will contain a

polynomial with xa+1p as nonzero summand. Thus neither of these subspaces can

be in the image of L1, and so must lie in the large stratum. Hence both these blocks

are missing.

Suppose now that the half-blocks happen to lie above our block labelled j. Then

L2u is either three- or four-dimensional. In the general case, it has dimension four

and there is a decomposition

L2u = u1− ⊕ u2

− ⊕ u+


with ui− both one-dimensional. It special situations L2u is only three dimensional,

and one of the ui−’s is missing (see Lemma 6.24 below for a detailed analysis). In

any case, xa+1p will lie in u+, forcing that subspace to be in the large stratum. The

other relevant polynomial xayp may or may not generate a one-dimensional invariant

subspace, depending on the values of a, p; so at least one, possibly both, of u1−, u

2−

lies in the image of L1 but not L21, forcing them to lie in the corresponding stratum.

So at least one, but possibly both, of the corresponding half-blocks must be missing.

We remark here that the other ui−, if present, can be divisible by a higher power

of L1. This implies that in this case the ideal generated by u may have nontrivial

intersection with the cells Vk,l,j even with l > 0.

The proofs of (3)-(4) follow the same pattern; we omit the details. Finally if I is

of finite codimension, then it contains Sm for m large enough, and so YI contains

only finitely many blocks and half-blocks. �

CHAPTER 6

Type Dn: decomposition of the orbifold Hilbert scheme

In this chapter we give a cell decomposition of the equivariant Hilbert scheme

for the type D case. This is based on a detailed analysis of the geometry of some

naturally defined incidence varieties. In some cases these incidence varieties happen

to have the structure of a join of two lower dimensional varieties.

6.1. The decomposition

The aim of this chapter is to prove the following result, which gives a constructive

proof of Theorem 3.2 for type Dn.

Theorem 6.1. Let G∆ be the subgroup of SL(2,C) of type Dn. Then there is a

locally closed decomposition

Hilb([C2/G∆]) =⊔

Y ∈Z∆

Hilb([C2/G∆])Y

of the equivariant Hilbert scheme Hilb([C2/G∆]) into strata indexed bijectively by the

set Z∆ of Young walls of type Dn, with each stratum Hilb([C2/G∆])Y a non-empty

affine space.

Proof. The affine plane C2 carries the diagonal T = C∗-action, which commutes

with the G∆-action. The action of T lifts to all the equivariant Hilbert schemes

Hilbρ([C2/G∆]) which are themselves nonsingular. Thus the fixed point set

Hilb([C2/G∆])T = tρHilbρ([C2/G∆])T

is also a union of nonsingular varieties, and it consists of points representing homo-

geneous invariant ideals. The construction of 5.6 associates a Young wall Y to each

homogeneous invariant ideal I � C[x, y]. Since the construction uses a locally closed

decomposition of the projective spaces Pm,j, the Young wall Y also depends in a

locally closed way on the ideal I, and thus we obtain a decomposition

Hilb([C2/G∆])T =⊔

Y ∈Z∆

ZY

into reduced locally closed subvarieties, where ZY ⊂ Hilb([C2/G∆])T is the locus of

homogeneous invariant ideals I with associated Young wall Y .

Let Hilb([C2/G])Y ⊂ Hilb([C2/G]) denote the locus of ideals which flow to ZYunder the action of the torus T . Then by the Bia lynicki-Birula theorem [4], there is

a regular map Hilb([C2/G])Y → ZY which is a Zariski locally trivial fibration with

53

54 6. TYPE DN : DECOMPOSITION OF THE ORBIFOLD HILBERT SCHEME

affine space fibres, and a compatible T -action on the fibres. By Theorem 6.3 below,

the base is an affine space as well. Hence, by [4, Sect.3, Remarks], Hilb([C2/G])Y is

an algebraic vector bundle over this base, and hence trivial (Serre–Quillen–Suslin)

[42]. Theorem 6.1 follows. �

Remark 6.2.

(1) As Hilb([C2/G∆]) = Hilb(C2)G∆ ⊂ Hilb(C2) is a smooth subvariety, the uni-

versal family over Hilb(C2) restricts to a universal family over the equivariant

Hilbert scheme Hilb([C2/G∆]). This restricts to a universal family of homo-

geneous invariant ideals U �OHilb([C2/G∆])T ⊗ C[x, y] over the T -fixed point

set. Restricting this universal family U to each of the strata constructed

above gives flat families of homogeneous invariant ideals UY �OZY ⊗C[x, y]

over each stratum ZY . It follows from the construction that the families UYare universal for flat families of homogeneous invariant ideals with associated

Young wall Y . We will have occasion to use the universal property of the

strata ZY below.

(2) The diagonal T = C∗-action on C2 induces the usual monomial grading

on C[x, y], and hence on each quotient C[x, y]/I for I homogeneous. For

an invariant ideal I this quotient carries a G∆ representation. Hence we

can define its multigraded Hilbert function. This means that there is a

Z-grading induced by the T = C∗-action and a grading (or rather labelling)

by the set {ρ0, . . . , ρn} associated to the G∆-action. By Lemma 5.6, the

multigraded Hilbert function of a homogeneous invariant ideal I �C[x, y] is

determined by its associated Young wall Y .

The following is the main technical result of this section.

Theorem 6.3. For each Y ∈ Z∆, the stratum ZY constructed above is isomorphic

to a nonempty affine space.

Remark 6.4. We note that our proof of Theorem 6.3 below certainly provides

some information about the dimension of the affine space ZY , and thus of the affine

space Hilb([C2/G∆])Y . We leave the study of these quantities, which could lead to

a refinement of Theorem 3.1 in the Grothendieck ring of varieties for type Dn, for

further study.

Our proof of Theorem 6.3, discussed below following some preparation, is a direct

inductive proof. We start with a series of examples which exhibit the range of issues

our proof will have to tackle; the discussions use results to be proved further below.

Throughout we take the simplest example n = 4, which exhibits all the nontrivial

features.

Example 6.5. Let Y1 be the triangle of size 3.

6.1. THE DECOMPOSITION 55

2

2

0

1

0

43

An invariant homogeneous ideal I corresponding to this Young wall necessarily

has a generator in V0,3,2 ⊂ P3,2. The latter is a projective line whose points can

be represented by expressions α0L2L3 + α1L2L4. The affine line V0,3,2 is given by

α0 + α1 6= 0. It is straightforward to check that a general point in V0,3,2, that is,

when [α0 : α1] 6∈ {[1 : 0], [0 : 1]}, indeed generates an ideal I with Young wall Y1.

However, when α0 or α1 become zero, then I does not intersect V1,3,4, respectively

V1,3,3, even though it should, so we have to add another generator to I from within

the corresponding cell (see Proposition 6.26(5) below). Both these cells are points, so

there is no further choice to make and thus the space ZY1 is isomorphic to an affine

line. This example already illustrates the fact that even within a single stratum ZY ,

the minimal number of generators of an ideal I with Young wall Y can vary.

Example 6.6. Let Y2 be the triangle of size 4. In this case, we get ZY2∼= ZY1

∼= A1,

the affine line of Example 6.5, due to the isomorphism V0,3,2∼= V0,4,0 × V0,4,1, see

Proposition 6.26(2) below, or repeat the same argument as above.

Example 6.7. Let Y3 be the triangle of size 5.

2 2

2

2

2

2

0

1

0

1

0

43

43

34

01

For each fixed ideal I with associated Young wall Y3 must necessarily have an

generator f ∈ I such that [f ] ∈ V0,5,2. By construction, V0,5,2 is an affine plane. This

generator is, up to scalar, unique, since the block of label 2 in position (0, 5) is the

only one with this label missing from the degree 5 antidiagonal. In other words,

I5,2 ∩ V0,5,2 should be a point, otherwise the points at infinity of I5,2 ∩ V0,5,2 would

intersect the other cells of degree 5 which is not allowed because of the shape of

the Young wall. I must also intersect both V1,5,0 and V1,5,1, and again in essentially

unique points, the corresponding blocks being the only 0/1 blocks missing from the

degree 6 antidiagonal. This puts the following constraint on the allowed [f ] ∈ V0,5,2.

Use the isomorphism L−11 which maps V1,5,0 t V1,5,1, a disjoint union of a point and

an affine line, to V0,4,1 t V0,4,0. Map this locus into V0,5,2, an affine plane, to obtain

M01 tM0

0 ⊂ V0,5,2 by taking the ideals generated by them (the curious notation

M01 tM0

0 for this locus is used here to be consistent with the general setup later,

see the definitions after Lemma 6.27). Then by Proposition 6.26(2) below, the ideal

〈f〉 intersects P6,0 and P6,1 at most in the correct cells V1,5,0 and V1,5,1 if and only


if [f ] ∈ V0,5,2 lies on the linear join of the affine line M00 and the point M0

1 inside

the affine plane V0,5,2 but not on M01 tM0

0 . This join is the plane V0,5,2 minus a

punctured affine line M1 \M01 , where M1 is the line parallel to M0

0 , going through

M01 . On this join, but away from the locus M0

0 tM01 itself, we can set I = 〈f〉 to

indeed get an ideal with Young wall Y3. On the locus M00 tM0

1 however, the ideal

〈f〉 itself will not actually meet both cells, so we have to add an arbitrary generator

of the missed cell to f to obtain an ideal of the correct Young wall. Over the affine

line M00 , there is no choice, since V1,5,0 is a point. But over the point M0

1 , we still

have V1,5,0, in other words an affine line, worth of choices. This sofar tells us that

ZY3 is the disjoint union of V0,5,2 \M1∼= A2 \ A1 and V1,5,0

∼= A1.

M01

M00M1

To fully work out the geometry of ZY3 , note that P5,2∼= P2 can be parameterized

by expersions α0L2L23 + α1L2L3L4 + α2L2L

24. The locus V0,5,2 ⊂ P5,2 is given by

α0 + α1 + α2 6= 0. The image of V0,4,0 in these coordinates is M00 = {(α0, 0, α1) :

α0 + α1 6= 0}, while the image of V0,4,1 is M01 = {(0, 1, 0)}. The linear combinations

of the points in M00 and M0

1 cover the whole affine plane V0,5,2, except a punctured

line. For a general linear combination a · (α0, 0, α1) + b · (0, 1, 0), the ideal generated

by the corresponding f intersects V1,5,0 × V1,5,1 in (L1L3L4, α0L1L23 + α1L1L

24). For

(a, b) = (1, 0) we have to have an extra generator in V1,5,0, while for (a, b) = (0, 1) we

need an extra generator in V1,5,1.

Consider a family of ideals which approaches the point M01 from the direction

(α0, 0, α1). Then it can be shown by explicit calculation that the limit ideal contains

the subspace generated by α0L1L23 + α1L1L

24 ∈ V1,5,1. This shows that ZY3 can be

obtained by blowing up the affine plane V0,5,2 in its point M01 , and removing the

proper transform of the punctured line M1 \M01 from this blowup. Thus ZY3

∼= A2.

From the blowup construction, we also obtain a canonical morphism ZY3 → A1, the

restriction of the morphism Bl0A2 → P1 to the exceptional curve of the blowup.

Example 6.8. Let Y ′3 be the Young wall

2

2

2

2

2

2

0

1

0

43

43

34

01

1

0

There is necessarily still a unique generator in V0,5,2. The difference compared to Y3

is that there is now no intersection with V1,5,1 but an intersection with V3,3,1. The

cells of the 0/1-blocks missing from the degree six diagonal are V1,5,0 and V3,3,1, both

6.2. INCIDENCE VARIETIES 57

of which are zero dimensional. As before, we pull back these using L−11 , take the

linear combinations of their images in P5,2, and intersect this line with V0,5,2. This is

exactly the line M1 from Example 6.7. This has one special point, M01 . If the new

generator is (in the subspace represented by) this point, then it will not generate an

ideal with shape Y , except if we keep the unique element of V3,3,0 as a generator. In

any case, ZY ′3∼= M1

∼= A1.

Example 6.9. It is well known that the minimal resolution of the singularity

C2/G∆ is given by the component Hilbρreg([C2/G]) of Hilb([C2/G]) corresponding to

the regular representation [40]. The C∗-fixed set on the minimal resolution consists

of the P1, the exceptional locus corresponding to the central node in the Dynkin

diagram, as well as three isolated points on the other three P1’s. The following five

Young walls contribute for the regular representation.

0 2 34 2 1 0 2 3

4 2

1

0 2 34

1 2

0 2 4

1 2 3

0 2 3

1 2 4

A quick computation shows that ZY is a point in each case, except for the last

Young wall in the first row, when it is an affine line similarly as in Example 6.5 but

in this case there is also a generator in V2,2,0. This affine line contains the point

corresponding to the Young wall next to it in its closure, giving the central P1.

6.2. Incidence varieties

The purpose of this section is to introduce some incidence varieties inside products

of the Schubert cells defined in 5.5. We state some propositions regarding these

incidence varieties and morphisms between them, whose proofs we defer to 6.6 below.

We discuss four different cases.

Case 6.2.1 Assume that m ≡ 0 mod n− 2 is a nonnegative integer, such that at

position (0,m) there is a divided block with labels (c1, c2). Let c be the label of

the block at position (1,m). Let Sc ⊆ Bm+1,c be a nonempty maximal subset. Let

S1 ⊆ Bm,c1 and S2 ⊆ Bm,c2 be two maximal subsets which are allowed by Sc. This

means, by definition, that each block above every composite block whose two halves

are both contained in S1 ∪ S2, and each block to the right of every composite block

at least one half-block of which in S1 ∪ S2, is in Sc.

Consider the incidence varieties

XScS1,S2

= {(U1, U2, Uc) : (U1, U2) ∩ Pm+1,c ⊆ Uc} ⊆ VS1 × VS2 × VSc,c,

and

Y ScS1,S2

= {(U1, U2, Uc) : (U1, U2) ∩ Pm+1,c ⊆ Uc} ⊆ VS1× VS2

× VSc,c.


These varieties fit into the diagram

XScS1,S2

⊆ VS1 × VS2 × VSc,cId×Id×ω−−−−−−−→ VS1 × VS2 × VSc,cyω×ω×Id

yω×ω×Id

Y ScS1,S2

⊆ VS1× VS2

× VSc,cId×Id×ω−−−−−−−→ VS1

× VS2× VSc,c .

Proposition 6.10.

(1) The image of XScS1,S2

under the vertical morphism VS1 × VS2 × VSc,c →VS1× VS2

× VSc,c is precisely Y ScS1,S2

.

(2) The induced morphism XScS1,S2

→ Y ScS1,S2

is a trivial fibration over its image

with affine fibers of dimension |Sc| − |S1| − |S2|+ 1.

(3) The horizontal morphism XScS1,S2

→ VS1 × VS2 × VSc,c is injective.

Case 6.2.2 Let m ≡ 0 mod n − 2, but this time consider only one half block of

label c0 = κ(c) at the position (0,m). Let Sc ⊆ Bm+2,c be a nonempty maximal

subset, and S ⊆ Bm,κ(c) be a maximal subset which is allowed by Sc. This means

that for each block in S there is a block in Sc at the top right corner. In analogy

with the previous case, let

XScS = {(U,Uc) : (U) ∩ Pm+2,c ⊆ Uc} ⊆ VS × VSc,c ,

and

Y ScS

= {(U,Uc) : (U) ∩ Pm+2,c ⊆ Uc} ⊆ VS × VSc,c ,

which fit into the diagram

XScS ⊆ VS × VSc,c

Id×ω−−−−−→ VS × VSc,cyω×Id

yω×Id

Y ScS⊆ VS × VSc,c

Id×ω−−−−−→ VS × VSc,c .

Proposition 6.11.

(1) The image of XScS under the vertical morphism VS × VSc,c → VS × VSc,c is

exactly Y ScS

.

(2) The induced morphism XScS → Y Sc

Sis a trivial fibration over its image with

affine fibers of dimension |Sc| − |S|.(3) The horizontal morphism XSc

S → VS × VSc,c is injective.

Case 6.2.3 Let m ≡ 1 mod n − 2, and c1 and c2 the labels of the divided block

immediately above the block at position (0,m). Let S1 ⊆ Bm+1,c1 , S2 ⊆ Bm+1,c2 be

nonempty subsets at least one of which is maximal. Let moreover, S ⊆ Bm,j be a

maximal subset which is allowed by S1 and S2. In this case, this means the following:

for each block b in S, there is a divided block of with labels (c1, c2) in the pattern

either immediately above or to the right of b. In the first case, we require that at

6.2. INCIDENCE VARIETIES 59

least one of these half-blocks is in S1 ∪ S2. In the second case, we require that both

are contained in S1 ∪ S2.

Given this data, we define

XS1,S2

S = {(U,U1, U2) : (U)∩Pm+1,c1 ⊆ U1, (U)∩Pm+1,c2 ⊆ U2} ⊆ VS×VS1,c1×VS2,c2 ,

and

Y S1,S2

S= {(U,U1, U2) : (U)∩Pm+1,c1 ⊆ U1, (U)∩Pm+1,c2 ⊆ U2} ⊆ VS×VS1,c1×VS2,c2 .

We now have the following diagram:

XS1,S2

S ⊆ VS × VS1,c1 × VS2,c2Id×ω×ω−−−−−−−→ VS × VS1,c1

× VS2,c2yω×Id×Id

yω×Id×Id

Y S1,S2

S⊆ VS × VS1,c1 × VS2,c2

Id×ω×ω−−−−−−−→ VS × VS1,c1× VS2,c2

.

Proposition 6.12.

(1) The image of XS1,S2

S under the vertical morphism VS × VS1,c1 × VS2,c2 →VS × VS1,c1 × VS2,c2 is exactly Y S1,S2

S.

(2) The induced morphism XS1,S2

S → Y S1,S2

Sis a trivial fibration with fibers

isomorphic to affine spaces of dimension |S1|+ |S2| − |S|.

We remark that the analogue of (3) of Propositions 6.10 and 6.11 is not true in

this case. What happens to XS1,S2

S when we project VS1,c1 × VS2,c2 to VS1,c1× VS2,c2

will be the subject of 7 below.

Case 6.2.4 Finally, assume m 6≡ 0, 1 mod n− 2 with a full block in position (0,m).

Let c be the label of the full block immediately above this position, and Sc ⊆ Bm+1,c

a nonempty maximal subset. Let moreover S ⊆ Bm,j be a maximal subset which is

allowed by Sc; in this case, this means that above every block of S there is a block

in Sc. Consider the incidence varieties

XScS = {(U,Uc) : (U) ∩ Pm+1,c ⊆ Uc} ⊆ VS × VSc,c

and

Y ScS

= {(U,Uc) : (U) ∩ Pm+1,c ⊆ Uc} ⊆ VS × VSc,c.

There is the following diagram:

XScS ⊆ VS × VSc,c

Id×ω−−−−−→ VS × VSc,cyω×Id

yω×Id

Y ScS⊆ VS × VSc,c

Id×ω−−−−−→ VS × VSc,c .

Proposition 6.13.

(1) The image of XScS under the vertical morphism VS × VSc,c → VS × VSc,c is

exactly Y ScS

.


(2) The induced morphism XScS → Y Sc

Sis a trivial fibration over its image with

affine fibers of dimension |Sc| − |S|.(3) The horizontal morphism XSc

S → VS × VSc,c is injective.

6.3. Proof of Theorem 6.3

In this section we prove Theorem 6.3, thus completing the proof of Theorem 6.1,

using the constructions and results stated in 6.2. Given a Young wall Y ∈ Z∆, we

need to show that the corresponding stratum ZY is an affine space. The following is

the key combinatorial definition which underlies much of the rest of the paper.

Definition 6.14. Consider the Young wall Y , as usual in the transformed pattern.

The salient blocks of Y are those blocks in the complement of Y , whose absence from

Y does not follow from the shape of the rows below it, and which are at the leftmost

positions in their rows with this property. In particular, these are

• missing half blocks under which there is a block in Y ;

• missing undivided full blocks under which there is a block in Y ;

• missing divided full blocks immediately to the right of the boundary of Y ;

• the leftmost missing block(s) in the bottom row.

Given an ideal I ∈ ZY , it is easy to see I is necessarily generated by elements

lying in cells corresponding to the salient blocks of Y . In most cases it is also true

that all cells corresponding to salient blocks must contain a generator, but not always;

we have already seen Example 6.5, where the divided missing blocks at position (1, 3)

are salient blocks of Y3, since they lie immediately to the right of the boundary of Y3,

but the corresponding cells do not necessarily contain generators of an ideal I ∈ ZY3 .

We start our analysis by defining maps from the strata ZY to the Grassmannian

cells defined in 5.5. Consider an arbitrary block or half-block of label j at position

(k, l) in the Young wall pattern. Let S(k, l, j) ⊆ Bk+l,j be the set of blocks of label

j at the positions (k′, l′) where k + l = k′ + l′, k′ ≥ k, and which are not in Y .

S(k, l, j) is called the index set of (k, l) in Y . If the block of label j at position

(k, l) is not contained in Y , then the index set S = S(k, l, j) contains (k, l) as well.

By the correspondence discussed at the end of Section 5.5 between the maximal

Schubert cells of the relevant Grassmannian for Vk,l,j and subsets of Bk+l,j(k) which

do contain (k, l), there is a maximal Schubert cell VS,j corresponding to S. The

affine cell VS,j ⊂ Grk+l,j for r = |S| parameterizes certain affine subspaces of Vk,l,j,

or, equivalently, projective subspaces of Pk+l,j, the projectivization of the space of

degree (k + l) homogeneous polynomials which transform in the representation ρjwith respect to G∆. The correspondence is by projective closure of the corresponding

affine subspace of Vk,l,j.

6.3. PROOF OF THEOREM 6.3 61

Lemma 6.15. For any block or half-block at position (k, l) which is not contained

in Y and has index set S = S(k, l, j), there is a morphism

ZY → VS,j

I 7→ I ∩ Vk,l,j.

Proof. Assume first for simplicity that (k, l) = (0,m). Let UY � (OZY ⊗ C[x, y])

be the universal family of homogeneous ideals over ZY introduced in Remark 6.2.

Consider the subfamily V = UY ∩ (OZY ⊗ Sm[ρj]). This is a family of subspaces of

dimension r dim ρj in Sm[ρj] parameterized by ZY . It is know, that the morphisms

from ZY to the equivariant Grassmannian Grm,j of Sm[ρj] are in one-to-one corre-

spondence with G∆ invariant subbundles of OZY ⊗ Sm[ρj ] of rank r dim ρj [13, p. 88,

Theorem 2.4]. Hence, there is a classifying morphism ZY → Grm,j inducing V .

By the definition of ZY , the multigraded Hilbert polynomial of UY is constant.

The Hilbert polynomial encodes the dimensions of the intersections with the cells

of Pm,j. Therefore, over closed points of ZY the elements of the family V, when

considered as projective subspaces of Pm,j, intersect exactly the cells Vki,li,j for

each element (ki, li) ∈ S. According to Section 5.5, these projective subspaces are

represented by points in the Schubert cell VS,j ⊂ Grm,j. Hence, the image of the

classifying morphism of V is inside the cell VS,j. By the construction, the classifying

morphism is just the same as taking the intersection of Im,j with V0,m,j. This is

denoted as I ∩ V0,m,j above.

The general case follows similarly as in Section 5.5. �

We will prove Theorem 6.1 by induction on the number of nonempty rows of

Y . Consider an arbitrary Young wall Y consisting of l > 0 rows. Let Y denote

the Young wall obtained from Y by deleting its bottom row; we will call this the

truncation of Y . Of course the labels of the half blocks are exchanged by κ, but we

will suppress this in the notations. The following result will be key to our induction.

Lemma 6.16. There exists a morphism of schemes

T : ZY → ZYI 7→ L−1

1 (I ∩ L1C[x, y]) .

Proof. Let UY � (OZY ⊗ C[x, y]) be the universal family of homogeneous ideals

over ZY . Consider I = L−11 (UY ∩ L1(OZY ⊗ C[x, y])). Is is straightforward to check

locally that this is still a sheaf of ideals in OZY ⊗ C[x, y]. On closed points of ZY ,

it is also clear that the restriction has Young wall Y . As mentioned above, the

multigraded Hilbert polynomial of UY is constant. As ZY is reduced, it then follows

from [31, Ch III. Thm. 9.9] that I is a flat family of homogeneous ideals with Young

wall Y over ZY . By Remark 6.2 there is a classifying morphism ZY → ZY for this

family, which is exactly the morphism T . �


Next, we continue with an investigation of the combinatorics of the bottom two

rows of our Young wall Y . The boundary of Y in the transformed pattern is divided

by the blocks into horizontal, vertical and diagonal straight line segments. The first

two lines in the bottom can be connected in the following six possible ways:

A1 A2 A3

B1 B2 B3

Here a diagonal straight line borders a half block of the Young wall, which can be

either a lower or an upper triangle. In the A cases the salient block in the bottom

row is a full block, while in the B cases it is a half block.

Let the salient block of Y in its bottom row be at position (0,m). It can be either

a divided or undivided full block, or a half block. In the first case, we have a type

A corner at the bottom of Y , while in the second case there is a type B corner. As

in 6.2, we need to consider four cases. In each case, we are going to define morphisms

ZY → XY and ZY → YY to incidence varieties defined in 6.2.

Case 6.3.1 Assume m ≡ 0 mod n− 2 and we have vertex types A1 or A2 (A3 is

not possible in this case). We are in the context of Case 6.2.1: the divided block at

position (0,m) has labels (c1, c2), and index sets S1, S2; the block at position (1,m)

has label c, and index set Sc. By the Young wall rules for Y , S1, S2 is allowed by Sc.

Lemma 6.15 implies that there is a morphism

ZY → VS1 × VS2 × VScI 7→ (I ∩ V0,m,c1 , I ∩ V0,m,c2 , I ∩ V1,m,c).

By construction, the image of this morphism is contained in the incidence variety

XScS1,S2

⊂ VS1 × VS2 × VSc from Case 6.2.1. Denote XY = XScS1,S2

⊆ VS1 × VS2 × VSc ;we thus obtain an induced morphism ZY → XY .

There is also a morphism

ZY → VS1× VS2

× VScI 7→ (L1I ∩ Vk1,l1,c1 , L1I ∩ Vk2,l2,c2 , L1I ∩ V1,m,c),

where (ki, li) is the lowest block in Si for i = 1, 2. We obtain an induced morphism

ZY → YY , where YY = Y ScS1,S2

.

Case 6.3.2 Assume m ≡ 0 mod n−2 with vertex types B1 to B3. This is Case 6.2.2:

the block at position (0,m) has label κ(c), and index set S = Sκ(c); the block at

position (1,m+ 1) has label c, and index set Sc. We consider the morphisms

ZY → VS × VScI 7→ (I ∩ V0,m,κ(c), I ∩ V1,m+1,c) ,


and

ZY → VS × VScI 7→ (L1I ∩ Vk,l,κ(c), L1I ∩ V1,m+1,c),

where again (k, l) is the lowest block in S. In this case we let XY = XScS and

YY = Y ScS

. The images of the morphisms above are contained in these.

Case 6.3.3 Assume m ≡ 1 mod n − 2 with vertex types A1, A2, A3. This is

Case 6.2.3: c1 and c2 are the labels of the divided block immediately above the block

at position (0,m), S1 ⊆ Bm+1,c1 , S2 ⊆ Bm+1,c2 are their index sets, both (in cases

A1 and A2) or one (in case A3) of which is maximal; S is the index set of the block

at position (0,m) with label j. We get a morphism

ZY → VS × VS1,c1 × VS2,c2

I 7→ (I ∩ V0,m,j, I ∩ V1,m,c1 , I ∩ V1,m,c2)

whose image is contained in XY = XS1,S2

S . In this way we obtain an induced morphism

ZY → XY,Y . Similarly, consider the morphism

ZY → VS × VS1,c1 × VS2,c2

I 7→ (L1I ∩ Vk,l,j, L1I ∩ V1,m,c1 , L1I ∩ V1,m,c2),

where (k, l) is the lowest block in S. We obtain an induced morphism ZY → YY ,

where YY = Y S1,S2

S.

Case 6.3.4 Assume finally that m 6≡ 1 mod n− 2 with vertex types A1 or A2. This

is Case 6.2.4: the full block at position (0,m) has label j and index set S which is

maximal; the full block at position (1,m) has label c and index set Sc; S is allowed

by Sc. We get a morphism

ZY → VS × VScI 7→ (I ∩ V0,m,j, I ∩ V1,m,c)

whose image is contained in XScS ⊆ VS × VSc , an incidence variety we denote by XY

to obtain an induced morphism ZY → XY,Y .

Second, let

ZY → VS × VScI 7→ (L1I ∩ Vk,l,j, L1I ∩ V1,m,c),

where (k, l) is the lowest block in S. By letting YY = Y ScS

we obtain an induced

morphism ZY → YY,Y .

The last key step in our inductive proof is the following result, valid in all four

cases above.

Proposition 6.17. The following is a scheme-theoretic fiber product diagram,

with the right hand vertical map in each case given by the map induced by statement


(1) of Propositions 6.10, 6.11 6.12 or 6.13 as appropriate.

(6.1)

ZY −−→ XYyT yω×Id

ZY −−→ YY .

Proof. It is immediate from the definitions in the different cases that the

diagram is commutative. We thus need to show that it is a fibre product. Let B

be an arbitrary base scheme and let f : B → ZY and g : B → XY be morphisms; we

need to show that these induce a unique morphism B → ZY . We consider Case 6.3.1;

the proof in the other cases is analogous. The map f corresponds to a flat family of

ideals If �OB ⊗ C[x, y] with Young wall Y . The map g corresponds to a 3-tuple of

families U1,g,U2,g,Uc,g of subspaces of C[x, y] over B. Given this data, consider the

family of ideals

If,g = (L1If ,U1,g,U2,g) �OB ⊗ C[x, y]

over B. Here there parentheses mean the generated ideal as explained in Section 6.5

below. By the compatibility of (f, g) it is immediate that the Young wall of the

corresponding ideals is Y . The classifying map of this family is the unique possible

extension of (f, g) to a morphism B → ZY . �

Conclusion of the Proof of Theorem 6.3. Assume that we have shown

for any Young wall Y having less then l rows that the corresponding stratum ZY is

affine, the l = 1 case being obvious. Consider an arbitrary Young wall Y consisting

of l rows. Let Y denote its truncation, as defined above. By the induction assumption,

the space ZY is affine. Also, by Propositions 6.10, 6.11, 6.12 or 6.13 respectively,

the map XY → YY of Proposition 6.17 is a trivial affine fibration in all cases. By

Proposition 6.17, the map ZY → ZY is a pullback of a trivial affine fibration and

thus itself a trivial affine fibration. Using the induction hypothesis, ZY is thus an

affine space. The proof of Theorem 6.3 is complete. �

Remark 6.18. One can deduce from the above proof that one can in fact

canonically choose generators of a homogeneous ideal I ∈ ZY , which are in the

cells of the some of the salient blocks of Y ; as discussed before, not all salient cells

necessarily contain a generator. For describing the coarse Hilbert scheme we have to

keep track of these generators, but we will do this only implicitly.

Example 6.19. Returning to Examples 6.6-6.7, we see that for Y3 the triangle

of side 5, Y3 = Y2, the triangle of size 4. The map T : ZY3∼= A2 → ZY3

∼= A1 is the

map identified at the end of the discussion of Example 6.7.

6.4. Digression: join of varieties

We now make a digression to a necessary construction from algebraic geometry.

6.5. PREPARATION FOR THE PROOF OF THE INCIDENCE PROPOSITIONS 65

Recall [1] that the join J(X, Y ) ⊂ Pn of two projective varieties X, Y ⊂ Pn is

the locus of points on all lines joining a point of X to a point of Y in the ambient

projective space. One well-known example of this construction is the following. Let

L1∼= Pk and L2

∼= Pn−k−1 be two disjoint projective linear subspaces of Pn.

Lemma 6.20. The join J(L1, L2) ⊂ Pn equals Pn. Moreover, the locus Pn\(L1∩L2)

is covered by lines uniquely: for every p ∈ Pn \ (L1 ∩ L2), there exists a unique line

P1 ∼= p1p2 ⊂ Pn with pi ∈ Li, containing p.

Let now H ⊂ Pn be a hyperplane not containing the Li, which we think of as

the hyperplane “at infinity”. Let V = Pn \ H ∼= An. Let Li = Li ∩ H, and let

Loi = Li \Li = Li∩V be the affine linear subspaces in V corresponding to Li. Finally

let X = J(L1, L2) ∼= Pn−1 and Xo = X \ (X ∩H) ∼= An−1.

Lemma 6.21. Projection away from L1 defines a morphism φ : X \ L1 → L2,

which is an affine fibration with fibres isomorphic to Ak. φ restricts to a morphism

φo : Xo → Lo2, which is a trivial affine fibration over Lo2∼= An−k−1 with the same

fibers isomorphic to Ak.

In geometric terms, the map φ is defined on X \ (L1 ∪ L2) as follows: take

p ∈ X \ (L1 ∪ L2), find the unique line p1p2 passing through it, with p1 ∈ L1 and

p2 ∈ L2; then φ(p) = p2.

Let now U be a projective subspace of H which avoids L2. Let U1 ⊂ U be a

codimension one linear subspace, and W = U \ U1 its affine complement. In the

main text, we need the following statement.

Lemma 6.22. χ((J(Lo2,W ) \ Lo2) ∩ V ) = 0.

Proof. With the same argument as in Lemma 6.21, J(Lo2,W )∩ V is a fibration

over Lo2 with fiber Cone(W ), and (J(Lo2,W ) \ Lo2) ∩ V is a fibration over Lo2 with

fiber Cone(W ) \ {vertex}. Since Cone(W ) \ {vertex} = C∗×W , the projection from

(J(Lo2,W )\Lo2)∩V ∼= Lo2×W ×C∗ to Lo2×W has fibers C∗. The lemma follows. �

The definition of join of varieties can be generalized for projective schemes

X, Y ⊂ PnS over an arbitrary base scheme S [1]. With the same arguments it can

be shown that this general definition satisfies too the properties mentioned above.

Moreover, it satisfies the following base-change property as well:

Lemma 6.23. [1, B1.2] Let S be an arbitrary scheme. Then for schemes X, Y ⊂PnS and an S-scheme T , we have the following equality in PnT :

J(X ×S T, Y ×S T ) = J(X, Y )×S T.

6.5. Preparation for the proof of the incidence propositions

To prepare the ground for the proof of the propositions announced in 6.2, con-

sider the operators defined in 5.4. We use these operators to describe projective


coordinates on some of the Grassmannians Pm,j. We first record the following equal-

ities, computing the isotypical summands of the homogeneous pieces of the ring

S = C[x, y].

Lemma 6.24. We have

S2k(n−2)[ρκ(0,k)] = (Ln−1 + Ln)2k[ρκ(0,k)] = (L2n−1 + L2

n)k;

S2k(n−2)[ρκ(1,k)] = (Ln−1 + Ln)2k[ρκ(1,k)] = Ln−1Ln(L2n−1 + L2

n)k−1;

S(2k+1)(n−2)[ρκ(n−1,k)] = (Ln−1 + Ln)2k+1[ρκ(n−1,k)] = Ln−1(L2n−1 + L2

n)k;

S(2k+1)(n−2)[ρκ(n,k)] = (Ln−1 + Ln)2k+1[ρκ(n,k)] = Ln(L2n−1 + L2

n)k.

Proof. For each equality on the right, use (5.1) and an easy induction argument.

For the equalities on the left, ⊇ is always clear; then use dimension counting. �

Thus, given an element v ∈ P2k(n−2),κ(0,k), we can write it uniquely in the form

v =k∑i=0

αiL2in−1L

2(k−i)n ,

for certain projective coordinates [α0 : · · · : αk]. Analogous coordinates exist on

P2k(n−2),κ(1,k) and P(2k+1)(n−2),j for j = n− 1, n.

For the two-dimensional representations we similarly have

Lemma 6.25. For 1 < j < n− 1,

S2k(n−2)+j−1[ρj] = Lj((Ln−1 + Ln)2k

)= Lj

((Ln−1 + Ln)2k[ρ0]

)⊕ Lj

((Ln−1 + Ln)2k[ρ1]

);

S(2k+2)(n−2)−j+1[ρj] = Ln−m((Ln−1 + Ln)2k+1

)= Ln−j

((Ln−1 + Ln)2k+1[ρn−1]

)⊕ Ln−j

((Ln−1 + Ln)2k+1[ρn]

).

Proof. Analogous. �

For such 1 < j < n−1, the space P2k(n−2)+j−1,j of two-dimensional G∆-equivariant

subspaces of S2k(n−2)+j−1[ρj ] is a projective space as noted above. Using Lemma 6.25,

we get a collection of distinguished two-dimensional G∆-equivariant subspaces

LjLin−1L

2k−in in S2k(n−2)+j−1[ρj]; an arbitrary element v ∈ P2k(n−2)+j−1,j can be

uniquely written as

v =2k∑i=0

αiLjLin−1L

2k−in

for certain projective coordinates [ε0 : . . . : ε2k]. Analogous coordinates also exist on

the space P(2k+2)(n−2)−j+1,j.

For subspaces U1, . . . , Ui ∈ Gr(S) denote by (U1, . . . , Ui) the G-invariant ideal of

S generated by the corresponding subspaces. In particular, the ideal generated by

(the subspaces represented by) points v1, . . . , vi ∈ PS is denoted by (v1, . . . , vi). Then


(U1, . . . , Ui)m,j is represented by a projective linear subspace of Pm,j (cf. Lemma 5.6).

Thus, we can talk about its intersection with the cells of Pm,j . For simplicity we will

use the notation (U1, . . . , Ui) ∩ Vk,l,j = (U1, . . . , Ui)k+l,j ∩ Vk,l,j for the intersection

with Vk,l,j.

We need to study indidence relations between ideals generated by subspaces

from the various strata defined above. First of all, let vj ∈ V0,m,j for some m and j,

corresponding to a full or half block. Denote by C the set of labels of full or half

blocks immediately above or on the right of this block, i.e. in the positions (1,m)

or (0,m+ 1). Then (vj) ∩ Sm+1,c = ∅ whenever c 6∈ C. Indeed, one has to analyze

the irreducible factors of L2vj as in the proof of Lemma 5.7. The following long

statement discusses all the remaining cases when c ∈ C. It splits according to the

different possibilities.

Proposition 6.26.

(1) For j = 0, 1, fix vj ∈ V0,2k(n−2),j.

(a) We have (vj) ∩(⋃

l≥1 Vl,2k(n−2)−l+1,2

)= ∅. Hence the unique point

(vj) ∩ P2k(n−2)+1 necessarily lies in V0,2k(n−2)+1,2. This provides an

injection

V0,2k(n−2),j → V0,2k(n−2)+1,2.

(b) (v0, v1) ∩(⋃

l>1 Vl,2k(n−2)−l+1,2

)= ∅. In particular, the projective line

(v0, v1) ∩ P2k(n−2)+1,2 necessarily intersects V1,2k(n−2),2. Let the inter-

section point be L1v2 for a certain v2 ∈ V0,2k(n−2)−1,2. Then v2 is

the unique point of V0,2k(n−2)−1,2 such that v0, v1 ∈ (v2). As a con-

sequence, for any projective subspace U2 ⊆ P2k(n−2)−1,2, the intersec-

tion (v0, v1) ∩(⋃

l>0 Vl,2k(n−2)−l+1,2

)is contained in L1U2 if and only if

v0, v1 ∈ (U2).

(2) Let v2 ∈ V0,2k(n−2)+1,2. For j = 0, 1, if (v2) ∩(⋃

l>0 Vl,2k(n−2)−l+2,j

)is not

empty, then it is necessarily one-dimensional, and it equals L1vj for a certain

vj ∈ V0,2k(n−2),j. Exactly one of the following three cases happens.

• (v2)∩(⋃

l>0 Vl,2k(n−2)−l+2,1

)= L1v0 and (v2)∩

(⋃l>0 Vl,2k(n−2)−l+2,0

)= ∅.

This happens if and only if v2 ∈ (v0). In this case, v0 ∈ V0,2k(n−2),0,

and (v2) ∩ S2k(n−2)+2 has two (resp. three if n = 4) irreducible com-

ponents: L1v0 and (v2) ∩ V0,2k(n−2)+2,3 (resp. (v2) ∩ V0,2k(n−2)+2,3 and

(v2) ∩ V0,2k(n−2)+2,4).

• (v2)∩(⋃

l>0 Vl,2k(n−2)−l+2,1

)= ∅ and (v2)∩

(⋃l>0 Vl,2k(n−2)−l+2,0

)= L1v1

with symmetrical statements as in the previous case.

• (v2) ∩(⋃

l>0 Vl,2k(n−2)−l+2,1

)= L1v0 and (v2) ∩

(⋃l>0 Vl,2k(n−2)−l+2,0

)=

L1v1. This happens if and only if v2 ∈ (v0, v1) but v2 /∈ (v0) ∪ (v1). In

this case at least one of the inclusions v0 ∈ V0,2k(n−2),0, v1 ∈ V0,2k(n−2),1

is satisfied but not necessarily both. Furthermore, (v2) ∩ S2k(n−2)+2 has


three (resp. four if n = 4) irreducible components: L1v0, L1v1 and

(v2) ∩ V0,2k(n−2)+2,3 (resp. (v2) ∩ V0,2k(n−2)+2,3 and (v2) ∩ V0,2k(n−2)+2,4).

Thus for n > 4, we obtain an isomorphism

V0,2k(n−2)+1,2 → V0,2k(n−2)+2,3

v2 7→ (v2) ∩ V0,2k(n−2)+2,3,

whereas for n = 4, we get an isomorphism

V0,2k(n−2)+1,2 → V0,2k(n−2)+2,3 × V0,2k(n−2)+2,4

v2 7→((v2) ∩ V0,2k(n−2)+2,3, (v2) ∩ V0,2k(n−2)+2,4

).

Finally, for projective subspaces U0 ⊆ P2k(n−2),0, U1 ⊆ P2k(n−2),1,

• the conditions (v2) ∩(⋃

l>0 Vl,2k(n−2)−l+2,1

)⊆ L1U0 and (v2) ∩(⋃

l>0 Vl,2k(n−2)−l+2,0

)= ∅ are satisfied if and only if v2 ∈ (U0);


l>0 Vl,2k(n−2)−l+2,1

)= ∅ and (v2) ∩(⋃

l>0 Vl,2k(n−2)−l+2,0

)⊆ L1U1 are satisfied if and only if v2 ∈ (U1);


l>0 Vl,2k(n−2)−l+2,1

)⊆ L1U0 and (v2) ∩(⋃

l>0 Vl,2k(n−2)−l+2,0

)⊆ L1U1 are satisfied if and only if v2 ∈ (U0, U1)

but v2 /∈ (U0) ∪ (U1).

(3) Assume that 3 ≤ j ≤ n − 3 (resp. j = n − 2) and set vj ∈ V0,2k(n−2)+j−1,j.

Then (vj) ∩(⋃

l>1 Vl,2k(n−2)−l+j,j−1

)= ∅. Furthermore, (vj) ∩ S2k(n−2)+j has

two (resp. three) irreducible components: a point (vj)∩V1,2k(n−2)+j−1,j−1 of the

form L1vj−1 for some vj−1 ∈ V0,2k(n−2)+j−2,j−1, which is the unique element

with vj ∈ (vj−1), and another point (vj) ∩ V0,2k(n−2)+j,j+1 (resp. two other

points (vj)∩V0,2k(n−2)+j,n−1, (vj)∩V0,2k(n−2)+j,n). We obtain an isomorphism

V0,2k(n−2)+j−1,j → V0,2k(n−2)+j,j+1

for j ≤ n− 3 and an isomorphism

V0,2k(n−2)+j−1,j → V0,2k(n−2)+j,n−1 × V0,2k(n−2)+j,n

for j = n− 2. Also, for a projective subspace Uj−1 ⊆ P2k(n−2)+j−2,j−1, the

intersection (vj)∩(⋃

l>0 Vl,2k(n−2)−l+j,j−1

)is contained in L1Uj−1 if and only

if vj ∈ (Uj−1).

(4) For j = n− 1, n, fix vj ∈ V0,(2k+1)(n−2),j.

(a) We have (vj)∩(⋃

l≥1 Vl,(2k+1)(n−2)−l+1,n−2

)= ∅. Hence, the point (vj)∩

P(2k+1)(n−2)+1 is necessarily in V0,(2k+1)(n−2)+1,n−2. This provides an

injection

V0,(2k+1)(n−2),j → V0,(2k+1)(n−2)+1,n−2.

(b) (vn−1, vn) ∩(⋃

l>1 Vl,(2k+1)(n−2)−l+1,2

)= ∅. In particular, the

projective line (vn−1, vn) ∩ P(2k+1)(n−2)+1,n−2 necessarily intersects

V1,(2k+1)(n−2),n−2. Let the intersection point be L1vn−2 for a cer-

tain vn−2 ∈ V0,(2k+1)(n−2)−1,n−2. Then vn−2 is the unique point of


V0,(2k+1)(n−2)−1,n−2 such that vn−1, vn ∈ (vn−2). As a consequence,

for any projective subspace Un−2 ⊆ P(2k+1)(n−2)−1,n−2, the intersection

(vn−1, vn) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+1,n−2

)is contained in L1Un−2 if and

only if vn−1, vn ∈ (Un−2).

(5) Let vn−2 ∈ V0,(2k+1)(n−2)+1,n−2. For j = n − 1, n if (vn−1) ∩(⋃l>0 Vl,(2k+1)(n−2)−l+2,j

)6= ∅ then this invariant subspace is one-

dimensional, and it is of the form L1vj for certain vj ∈ V0,(2k+1)(n−2),j.

Exactly one of the following three possibilities happens.

• (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n

)= L1vn−1 and (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n−1

)= ∅. This happens if and only

if vn−2 ∈ (vn−1). In this case vn−1 ∈ V0,(2k+1)(n−2),n−1, and

(v2)∩S(2k+1)(n−2)+2 has two (resp. three if n = 4) irreducible components:

L1vn−1 and (vn−2) ∩ V0,(2k+1)(n−2)+2,n−3 (resp. (v2) ∩ V0,(2k+1)(n−2)+2,0

and (v2) ∩ V0,(2k+1)(n−2)+2,1).

• (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n

)= ∅ and (vn−2) ∩(⋃

l>0 Vl,2k(n−2)−l+2,n−1

)= L1vn with symmetrical statements as

in the previous case.

• (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n

)= L1vn−1 and (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n−1

)= L1vn. This happens if and only if

vn−2 ∈ (vn−1, vn) but vn−2 /∈ (vn−1) ∪ (vn). In this case at least one of

the inclusions vn−1 ∈ V0,(2k+1)(n−2),n−1, vn ∈ V0,(2k+1)(n−2),n is satisfied

but not necessarily both. Furthermore, (vn−2) ∩ S(2k+1)(n−2)+2 has

three (resp. four if n = 4) irreducible components: L1vn−1, L1vnand (vn−2) ∩ V0,(2k+1)(n−2)+2,n−3 (resp. (vn−2) ∩ V0,(2k+1)(n−2)+2,0 and

(vn−2) ∩ V0,(2k+1)(n−2)+2,1).

For n > 4, we obtain isomorphisms

V0,(2k+1)(n−2)+1,n−2 → V0,(2k+1)(n−2)+2,n−3

vn−2 7→ (vn−2) ∩ V0,(2k+1)(n−2)+2,n−3

,

whereas for n = 4 we obtain an isomorphism

V0,(2k+1)(n−2)+1,n−2 → V0,2k(n−2)+2,0 × V0,(2k+1)(n−2)+2,1

vn−2 7→((vn−2) ∩ V0,(2k+1)(n−2)+2,0, (vn−2) ∩ V0,(2k+1)(n−2)+2,1

) .Moreover, for projective subspaces Un−1 ⊆ P(2k+1)(n−2),n−1, Un ⊆P(2k+1)(n−2),n the conditions

• (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n

)⊆ L1Un−1 and (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n−1

)= ∅ are satisfied if and only if

vn−2 ∈ (Un−1);

• (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n

)= ∅ and (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n−1

)⊆ L1Un are satisfied if and only if

vn−2 ∈ (Un);


• (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n

)⊆ L1Un−1 and (vn−2) ∩(⋃

l>0 Vl,(2k+1)(n−2)−l+2,n−1

)⊆ L1Un are satisfied if and only if

vn−2 ∈ (Un−1, Un) but v2 /∈ (Un−1) ∪ (Un).

(6) Assume that 3 ≤ j ≤ n − 3 (resp. j = 2) and set vj ∈V0,(2k+2)(n−2)−j+1,j. Then (vj) ∩

(⋃l>1 Vl,(2k+2)(n−2)−l−j+2,j+1

)= ∅. Fur-

thermore, (vj) ∩ S(2k+2)(n−2)−j+2 has two (resp. three) irreducible compo-

nents: a point (vj) ∩ V1,(2k+2)(n−2)−j+1,j+1 of the form L1vj+1 for some

vj+1 ∈ V0,(2k+2)(n−2)−j,j+1, which is the unique element with vj ∈ (vj+1),

and another point (vj) ∩ V0,(2k+2)(n−2)−j+2,j−1 (resp. two other points

(vj) ∩ V0,(2k+2)(n−2)−j+2,0, (vj) ∩ V0,(2k+2)(n−2)−j+2,1) providing an isomor-

phism V0,(2k+2)(n−2)−j+1,j → V0,2k(n−2)−j+2,j−1 (resp. V0,(2k+2)(n−2)−j+1,j →V0,2k(n−2)−j+2,0×V0,2k(n−2)−j+2,1). As a consequence, for a projective subspace

Uj+1 ⊆ P(2k+2)(n−2)−j,j+1, the intersection (vj)∩(⋃

l>0 Vl,(2k+2)(n−2)−l−j+2,j+1

)is contained in L1Uj+1 if and only if vj ∈ (Uj+1).

Proof. For ease of notation in the proof, we will assume that n is even. If n is

odd, then the argument works in the same way, except that κ should be applied to

the indices when appropriate.

(1) Statement (a) follows immediately from the definition of the cells. For the

first part of (b), write v0 =∑k

i=0 αiL2in−1L

2(k−i)n and v1 =

∑k−1i=0 βiL

2i+1n−1 L

2(k−i)−1n . The

assumptions guarantee that∑αi 6= 0 and

∑βi 6= 0. The image (v0, v1)∩P2k(n−2)+1,2,

as subset of Gr(2, S2k(n−2)+1[ρ2]), is a projective line and is spanned by

L2v0 =k∑i=0

αiL2L2in−1L

2(k−i)n ∈ L2(L2

n−1 + L2n)k

and

L2v1 =k−1∑i=0

βiL2L2i+1n−1 L

2(k−i)−1n ∈ L2Ln−1Ln(L2

n−1 + L2n)k−1.

In particular, if (v0, v1) ∩ V1,2k(n−2) = {L1v2}, then there exist vectors vx, vy in the

two-dimensional vector space v2 satisfying vy = τ(vx), as well as ax, bx, ay, by ∈ C, so

that

(6.2)L1vx = axL2,xv0 + bxL2,xv1,

L1vy = ayL2,yv0 + byL2,yv1.

In v0 the highest powers of x and y are x2k(n−2) and y2k(n−2), both with coefficient∑i αi. In v1 the highest powers of x and y are x2k(n−2) and y2k(n−2), the first

has coefficient∑

i βi, the second has coefficient −∑

i βi. We apply L2,x to these.

In order for the sum to avoid the cell V0,2k(n−2)+1, the coefficients must satisfy

[ax : bx] = [∑

i βi : −∑

i αi], since in this case the coefficient of x2k(n−2)+1 is 0. Then

the linear combination is necessarily in V1,2k(n−2). Similarly, when applying L2,y,

the required coefficients are [ay : by] = [∑

i βi :∑

i αi], so we have axby = −aybx.


Therefore, byL2,yL1vx − bxL2,xL1vy = byaxL2,yL2,xv0 − bxayL2,xL2,yv0 = L1v0, where

we have used that L2,xL2,y = L1. As a consequence, byL2,yvx + bxL2,xvy = v0 and

similarly −ayL2,yvx + axL2,xvy = v1, i.e. v0, v1 ∈ (v2). This proves the first part of

(b). The second part of (b), concerning projective subspaces, follows immediately

from the first part.

(2) For the first part of the statement let v2 =∑2k

i=0 εiL2Lin−1L

2k−in =

L2

(∑2ki=0,i even εiL

in−1L

2k−in +

∑2ki=0,i odd εiL

in−1L

2k−in

)= L2(veven + vodd). If n 6= 4,

then by applying L2 again we get L22(veven + vodd) = (L1 + L3)(veven + vodd), where

the first sum is operator sum, and the second is vector sum. So L2v2[ρ0] =

{L1veven} and L2v2[ρ1] = {L1vodd}. If (v2) ∩(⋃

l>1 Vl,2k(n−2)−l+2,1

)= L1v0 and

(v2) ∩(⋃

l>1 Vl,2k(n−2)−l+2,0

)= ∅, then veven = v0 and vodd = 0. The second

case is just the opposite, and when (v2) ∩(⋃

l>1 Vl,2k(n−2)−l+2,1

)= L1v0 and

(v2) ∩(⋃

l>1 Vl,2k(n−2)−l+2,0

)= L1v1, then veven = av0 and vodd = bv1 for some

coefficients a, b ∈ C. If n = 4, then L22(veven + vodd) = (L1 + L3 + L4)(veven + vodd),

and the rest is very similar to the first case.

For the second part of the statement observe that the results so far imply that

V0,2k(n−2)+1,2 stratifies into disjoint, locally closed subspaces

V0,2k(n−2)+1,2 =⊔

(vc1 ,vc2 )∈P2k(n−2),c1×P2k(n−2),c2

((vc1 , vc2) \ ((vc1) ∪ (vc2)) ∩ V0,2k(n−2)+1,2

)⊔ ⊔

vc2∈V0,2k(n−2),c2

((vc2) ∩ V0,2k(n−2)+1,2

)⊔ ⊔

vc1∈V0,2k(n−2),c1

((vc1) ∩ V0,2k(n−2)+1,2

).

The subset ∪(vc1 ,vc2 )∈V0,2k(n−2),c1×V0,2k(n−2),c2

((vc1 , vc2) \ ((vc1) ∪ (vc2)) ∩ V0,2k(n−2)+1,2

)is dense in the third stratum, since V0,2k(n−2),c1 × V0,2k(n−2),c2 is dense in P2k(n−2),c1 ×P2k(n−2),c2 . The first statement of (2) implies that the second statement is

valid if v2 is in this subset of V0,2k(n−2)+1,2. Similarly, the first statement im-

plies the second statement on the loci tvc1∈V0,2k(n−2),c1

((vc1) ∩ V0,2k(n−2)+1,2

)and

tvc2∈V0,2k(n−2),c2

((vc2) ∩ V0,2k(n−2)+1,2

). For v2 in the closed complement of the union

of these loci, the second statement follows from the linearity (and thus continuity) of

the solution of (6.2), since the Ui are projective.

(3) The statements in this case follow similarly to (2) by observing that L2Lj =

L1Lj−1 + Lj+1 (resp. L2Ln−2 = L1Ln−3 + Ln−1 + Ln).

The cases (4), (5) and (6) are analogous. �

Consider a full block in position (0,m) with m = k(n− 2) + 1, with label j which

is 2 or n− 2. In positions (0, k(n− 2)) and (1,m), above and to the left of this full

block, are divided blocks with labels (c1, c2), either (0, 1) or (n − 1, n). The next

lemma gives Pm,j the structure of a join of two projective subspaces.


Lemma 6.27.

(1) The morphisms V0,k(n−2),ci → V0,m,j constructed in Proposition 6.26 extend

to morphisms

φi : Pk(n−2),ci → Pm,jv 7→ (v) ∩ Pm,j.

The morphism φi is injective with image N0ci

:= im(φi) ⊂ Pm,j such that N0c1

,

N0c2

are disjoint projective linear subspaces of Pm,j.

(2) The join of the disjoint linear subspaces N0c1, N0

c1⊂ Pm,j is Pm,j itself.

Thus given (v1, v2) ∈ Pk(n−2),c1 × Pk(n−2),c2, there is a projective line P1 ∼=v1v2 ⊂ Pm,j containing both φi(vi), namely, the line defined by v1, v2 with

endpoints φi(vi). The lines v1v2 cover Pm,j. For all (v1, v2), (v′1, v′2) ∈

Pk(n−2),c1 × Pk(n−2),c2, the intersection v1v2 ∩ v′1v′2 can only be at a common

endpoint.

(3) For all (v1, v2) ∈ Pk(n−2),c1 × Pk(n−2),c2, the intersection v1v2 ∩ V0,m,j is

• either empty, exactly when v1 /∈ V0,k(n−2),c1 and v2 /∈ V0,k(n−2),c2;

• or an affine line otherwise.

Proof. (1) is immediate. (2) then follows from dimPk(n−2),c1+dimPk(n−2),ci+1 =

dimPm,j and Lemma 6.20. (3) is again immediate. �

As we did already in the statement above, we will sometimes omit the inclusion

maps φi; thus, for subspaces U1 ⊆ Pk(n−2),c1 and U2 ⊆ Pk(n−2),c2 , we will denote by

J(U1, U2) ⊆ Pm,j the join of φ1(U1) and φ1(U1) in Pm,j.

Let

M0ci

:= φi(V0,k(n−2),ci) ⊂ V0,m,j;

these are disjoint affine linear subspaces of the affine space V0,m,j. Also consider

Nci = J(Pk(n−2),ci , P k(n−2),c3−i) ⊂ Pm,j.

This is the locus of points in Pm,j covered by lines v1v2 one of whose endpoints is at

a point “at infinity”, in P k(n−2),c3−i = Pk(n−2),c3−i \ V0,k(n−2),c3−i . Let

Mci = Nci ∩ V0,m,j ⊂ V0,m,j

be the intersection with the large affine cell of Pm,j.

Lemma 6.28. There exists morphisms ψi : Mci →M0ci

, given by associating to a

point

v ∈Mci ⊂ V0,m,j

the “non-infinity” endpoint of the (mostly unique) line v1v2 passing through it. The

maps ψi are trivial vector bundles over affine spaces.

Proof. See Lemma 6.21. �

Corollary 6.29.

6.6. PROOFS OF PROPOSITIONS ABOUT INCIDENCE VARETIES 73

(1) For i = 1, 2 the decomposition Pk(n−2),c3−i =⊔

(k′,l′)∈Bk(n−2),c3−iVk′,l′,c3−i in-

duces a decomposition into locally closed subspaces

(6.3) Mci \M0ci

=⊔

(k′,l′)∈Bk(n−2),c3−i\(1,m)

((J(V0,k(n−2),ci , Vk′,l′,c3−i) ∩ V0,m,j) \M0

ci

).

(2) Taking into account the bijections Bk(n−2),ci∼= Bk(n−2)+2,c3−i and the de-

composition (6.3), the space V0,m,j decomposes into locally closed subspaces

asV0,m,j =V0,m,j(1,m, 1,m)

⊔ ⊔(k1,l1)∈B′

k(n−2)+2,c1

V0,m,j(k1, l1, 1,m)

⊔ ⊔

(k2,l2)∈B′k(n−2)+2,c2

V0,m,j(1,m, k2, l2)

,

where we introduced the notations

• B′k(n−2)+2,ci= (Bk(n−2)+2,ci ∪ {∅}) \ {(1,m)};

• V0,m,j(∅, 1,m) = M0c1

;

• V0,m,j(k1, l1, 1,m) = (J(V0,k(n−2),c1 , Vk1,l1,c2) ∩ V0,m,j) \M0c1

;

• V0,m,j(1,m, ∅) = M0c2

;

• V0,m,j(1,m, k2, l2) = (J(V0,k(n−2),c2 , Vk2,l2,c1) ∩ V0,m,j) \M0c2

;

• V0,m,j(1,m, 1,m) = V0,m,j \ (Mc1 ∪Mc2).

The meaning of the notations is that V0,m,j(k1, l1, k2, l2) consists of exactly those

points v ∈ V0,m,j such that (v) ∩ Pk(n−2)+2,ci ∈ Vki,li,ci for i = 1, 2. The symbol ∅ at

an argument replacing a pair (ki, li) means that there is no such intersection.

6.6. Proofs of propositions about incidence vareties

Here we prove the propositions announced in 6.2. The arguments for Propositions

6.10, 6.11 and 6.13 are very similar, so we will spell out the proof for one of these.

The discussion will also prepare the ground for the proof of Proposition 6.12, which

is substantially more complicated.

Consider first the situation of 6.13. Namely, m 6≡ 0, 1 mod n − 2 is a positive

integer, V0,m,j the cell of a full block, c is the label of the full block immediately

above the position (0,m), Sc ⊆ Bm+1,c is a nonempty maximal subset, and S ⊆ Bm,j

is a maximal subset which is allowed by Sc.

Proof of Proposition 6.13. By Proposition 6.26 (3) and (6), for an arbitrary

U ∈ VS, (U,Uc) ∈ XScS if and only if U ⊆ (L−1

1 Uc)∩V0,m,j . Moreover, the composition

of L−11 and the isomorphism V0,m−1,c → V0,m,j gives an isomorphism V1,m,c → V0,m,j.

Hence, for a pair (U,Uc) ∈ Y ScS

, we have

{U ∈ VS|U : (U,Uc) ∈ XScS } =

((L−1

1 Uc) ∩ V0,m,j

)/U,


and there is a canonical quotient map V0,m,j/U → V1,m,c/Uc.

Let us define two families parameterized by VS × VSc,c. The family Fc is defined

to have the fiber (L−11 Uc)∩ V0,m,j over a pair (U,Uc) ∈ VS × VSc,c. This is a family of

affine subspaces of V0,m,j . The family F is defined to have the fiber U ⊂ Pm,j over a

pair (U,Uc) ∈ VS × VSc,c. This is a family of projective subspaces contained in Pm,j.

Since the tautological bundle over any Schubert cell in any Grassmannian is trivial,

the two families are trivial with affine, respectively projective space fibres. Consider

these families over the subset Y ScS⊂ VS × VSc,c. By construction, over each point of

Y ScS

, the fibre of F is a subspace of the projective closure of the fiber of Fc over the

same point. In particular, we can take quotients fiberwise. By the considerations

above,

XS1,S2

S = Fc/F .

Moreover, the morphism ω × Id : XS1,S2

S → Y ScS

over a pair (U,Uc) is given by the

quotient morphism V0,m,j/U → V1,m,c/Uc times the identity. This shows (1) and (2).

The injectivity statement (3) follows again from the isomorphism V1,m,c∼= V0,m,j

given by L1, since for every pair (U,Uc) ∈ XS1,S2

S one has Uc = (U,U c) ∩ V1,m,c. �

Consider now the situation of Proposition 6.12; thus m ≡ 1 mod n − 2 is a

positive integer, c1 and c2 are the labels of the divided block immediately above the

block at position (m, j), S1 ⊆ Bm+1,c1 , S2 ⊆ Bm+1,c2 are nonempty subsets at least

one of which is maximal, and S ⊆ Bm,j is a maximal subset, which is allowed by S1

and S2.

Lemma 6.30. For i = 1, 2 fix Ui ∈ VSi,ci.

(a) For an arbitrary U ∈ VS, (U,U1, U2) ∈ XS1,S2

S if and only if U ⊆J(φ1(L−1

1 U1), φ2(L−11 U2)) ∩ V0,m,j.

(b) If (U,U1, U2) ∈ Y S1,S2

S, then U ⊆ J(φ1(L−1

1 U1), φ2(L−11 U2)) ∩ V0,m,j.

(c) If (U,U1, U2) ∈ Y S1,S2

S, then

{U ∈ VS|U : (U,U1, U2) ∈ XS1,S2

S } =(J(φ1(L−1

1 U1), φ2(L−11 U2)) ∩ V0,m,j

)/U.

Proof. (a) By Proposition 6.26 for any pair of vectors (v1, v2) ∈ U1 × U2 those

points of Pm,j for which (v1, v2) ∩ Pm+1,ci is either vi or empty are exactly those

which are on J(φ1(L−11 v1), φ2(L−1

1 v2)). Hence, to satisfy the conditions U has to be

a subset of⋃(v1,v2)∈U1×U2

J(φ1(L−11 v1), φ2(L−1

1 v2)) ∩ V0,m,j = J(φ1(L−11 U1), φ2(L−1

1 U2)) ∩ V0,m,j.

(b) If (U,U1, U2) ∈ Y , then (U)∩ Pm+1,ci ⊆ Ui. Hence, φi(L−11 ((U)∩ Pm+1,ci)) ⊆

φi(L−11 Ui), and

J(φ1(L−11 ((U) ∩ Pm+1,c1)), φ2(L−1

1 ((U) ∩ Pm+1,c2))) ⊆ J(φ1(L−11 U1), φ2(L−1

1 U2)).

6.6. PROOFS OF PROPOSITIONS ABOUT INCIDENCE VARETIES 75

By Proposition 6.26 there is an isomorphism V1,m−1,j∼= V0,m−1,c1 × V0,m−1,c1 in such

a way that

U ∩ V1,m−1,j ⊆ J(φ1(L−11 ((U) ∩ Pm+1,c1)), φ2(L−1

1 ((U) ∩ Pm+1,c2))) ∩ V1,m−1,j.

Similarly, on each cell Vk,l,j such that k + l = m and k ≥ 1, the affine subspace

U ∩ Vk,l,j is a subvariety of J(φ1(L−11 ((U)∩Pm+1,c1)), φ2(L−1

1 ((U)∩ Vm+1,c2)))∩ Vk,l,j.All these mean that U ⊆ J(φ1(L−1

1 U1), φ2(L−11 U2)).

(c) Recall, that U also represents a subspace at infinity for V0,m,j, and VS|U =

V0,m,j/U . In fact, we can take the quotient of an arbitrary subspace of V0,m,j, whose

closure in Pm,j contains U with respect to (an arbitrary affine subspace representing)

U . Then the statement follows from (a) and (b). �

Proof of Proposition 6.12. It follows from the definitions that

(ω × Id× Id) (XS1,S2

S ) ⊆ Y S1,S2

S. The surjectivity will follow from the calcula-

tion of the fibers.

We will define three families of subspaces in Pm,j over VS × VS1,c1 × VS2,c2 . For

i = 1, 2 the family Fi is defined to have the fiber φi(L−11 Ui) ⊆ Pm,j over a three-tuple

(U,U1, U2) ∈ VS × VS1,c1 × VS2,c2 . Let the third family F has the fiber U ⊆ Pm,j over

the same element. This is of course empty, if |S| = 1. It is important to note, that

in all cases the fibers are always projective subspaces of Pm,j.

By Lemma 6.27, there is an embedding φi ◦L−11 : Pm+1,c3−i → N0

ci⊂ Pm,j . Apply

this embedding on the fibers of the projectivization of the tautological bundle over

the Schubert cell VS3−i . Then multiply the base with VS ×VSi , and extend the family

into this direction as a constant. This gives the bundle Fi. Again, by the fact that

the tautological bundle over any Schubert cell is trivial it follows that the Fi’s are also

trivial, that is, Fi ∼= P|Si|−1× VS × VS1 × VS2 . Similarly, F ∼= P|S|−2× VS × VS1 × VS2 .

By Lemma 6.23, the join of trivial families over a common base is a trivial family

of the joins of the fibers:

J(F1,F2) = J(P|S1|−1 × VS × VS1 × VS2 ,P|S2|−1 × VS × VS1 × VS2)

∼= J(P|S1|−1,P|S2|−1)× VS × VS1 × VS2

∼= P|S1|+|S2|−1 × VS × VS1 × VS2 ⊆ Pm,j × VS × VS1 × VS2 .

Therefore, J(F1,F2)∩ (V0,m,j × VS × VS1 × VS2) is a trivial family of affine subspaces

of V0,m,j over VS × VS1 × VS2 .

By Lemma 6.30 (b) F is a (trivial) subfamily of J(F1,F2) over Y S1,S2

S. By Lemma

6.30 (c) XS1,S2

S can be constructed as

XS1,S2

S = (J(F1,F2) ∩ (V0,m,j × VS × VS1 × VS2))/F|YS1,S2S

.

Hence, XS1,S2

S is a trivial family of affine spaces of dimension |S1|+ |S2|− |S|, since it

is the quotient of a trivial affine family of fibre dimension |S1|+ |S2| − 1 by another

trivial affine family of fibre dimension |S| − 1. �

CHAPTER 7

Type Dn: special loci

In this chapter we analyze those special points of the equivariant Hilbert scheme

where a salient block of the associated Young wall fails to contain a generator of

the corresponding ideal. More precisely, the case we are interested in is when the

intersection of an ideal with a salient cell of its Young wall is already generated by

the subspaces of the ideal sitting in cells in lower rows. For this we will describe the

special points on the equivariant Grassmannian and then on the incidence varieties.

7.1. Support blocks

We need to analyze the cases when a cell corresponding to a salient block (see

Def. 6.14) of a Young wall Y fails to contain a generator of a corresponding ideal

I ∈ ZY . As an example, recall once again Example 6.5, where the divided missing

blocks at position (1, 3) are salient blocks of Y3, but the corresponding cells do not

necessarily contain generators of an ideal I ∈ ZY3 , i.e. they are already generated

by the “parts” of I which are in the cells of the blocks in the lower rows. That

this phenomenon can happen at all is one of the main sources of difficulty in our

analysis of the strata of the singular Hilbert scheme. We will introduce the notion of

a support block for a salient block below. Intuitively, the intersection of an ideal I

with the cell corresponding to the support block can generate the affine subspace

given by the intersection of I with the cell corresponding to the salient block (such

as the support block at position (0, 3) for Y3), and thus the salient block contains

no new generator of I. We will make this statement more precise in the rest of this

chapter.

We start with some combinatorial preliminaries. Recall the setup of Proposi-

tion 6.12: m ≡ 1 mod n−2; c1 and c2 are the labels of the divided block immediately

above the block of label j at position (m, 0); S1 ⊆ Bm+1,c1 and S2 ⊆ Bm+1,c2 are

nonempty subsets at least one of which is maximal; S ⊆ Bm,j is a maximal subset

which is allowed by S1 and S2.

For a half-block b of Si, consider the following two conditions.

(1) The blocks below or to the left of b are not contained in S.

(2) The block below b is contained in S, the complementary half-block b′ is

contained in S3−i, and the block to the left of their position in not contained

in S.

77

78 7. TYPE DN : SPECIAL LOCI

For i, j = 1, 2, let us denote by Ss,ji ⊂ Si the subset of half-blocks of label ci satisfying

condition (j). Let moreover Ssi = Ss,1i ∪ Ss,2i .

The next lemma, whose proof is immediate, connects the global Definition 6.14

for a Young wall Y with the local conditions (1)-(2) where we consider only the m-th

and m+ 1-st diagonals for a particular m, and index sets S, S1 and S2 as above.

Lemma 7.1. Given a Young wall Y ∈ Z∆, let S, respectively S1 and S2 denote

the set of missing blocks, respectively half-blocks of Y on the m-th and (m + 1)-st

diagonals. The blocks Ssi ⊂ Si are exactly the salient blocks of Y of label ci on the

(m+ 1)-st diagonal.

Thus we can legitimately call the blocks in Ssi salient blocks in this local situation.

Let us introduce the following subsets of S.

• Sl consists of blocks b ∈ S that are directly to the left of a divided block

with labels (c1, c2).

• Sb,0 consists of blocks b ∈ S, so that b is immediately below a divided block

with labels (c1, c2), the block immediately up-left of b is not in S, and both

of the divided blocks above b are in S1 ∪ S2.

• Sb,ci consists of blocks b ∈ S that are immediately below a divided block, so

that the block immediately up-left of b is not in S, and the block of label

c3−i above b is in S3−i.

• Sb,c1∪c2 consists of blocks b ∈ S such that b lies immediately below a divided

block, and the block immediately up-left of b is contained in S.

• Sb = Sb,0 ∪ Sb,c1 ∪ Sb,c2 ∪ Sb,c1∪c2 .

Note that by the Young wall rules, we necessarily have Sb = S \ Sl. We will call the

blocks in the set Sci = Sb,0 ∪ Sb,ci ∪ Sb,c1∪c2 support blocks for label ci. We will define

a support relation from Sci to Ssi in the next section.

7.2. Special loci in orbifold strata and the supporting rules

Let Y ∈ Z∆ be a Young wall with a salient block in its bottom row in position

(0,m) with m ≡ 1 mod n− 2, a full block immediately below a divided block with

labels (c1, c2). As before, let S, respectively S1 and S2 denote the set of missing

blocks, respectively half-blocks of Y on the m-th and (m+ 1)-st diagonals with the

corresponding labels.

We introduce index sets depending on S, S1 and S2. We consider two cases.

If S is not maximal, then let

I(S, S1, S2) =

{(k1, l1, k2, l2) :

(ki, li) ∈ Ssi ∪ {∅} for i = 1, 2,

and at least one (ki, li) = (1,m)

}.

We partition this index set into the following (possibly empty) disjoint subsets:

7.2. SPECIAL LOCI IN ORBIFOLD STRATA AND THE SUPPORTING RULES 79

• I(S, S1, S2)0 = {(k1, l1, k2, l2) ∈ I(S, S1, S2) : (ki, li) /∈{∅, (1,m)} for some i = 1, 2};• I(S, S1, S2)1 = {(1,m, ∅), (∅, 1,m)} ∩ I(S, S1, S2);

• I(S, S1, S2)−1 = {(1,m, 1,m)} ∩ I(S, S1, S2).

If S is maximal, then let

I(S, S1, S2) = {(k1, l1, k2, l2) : (ki, li) ∈ Ssi ∪ {∅} for i = 1, 2}.

We remark that in this case (1,m) /∈ Ssi for both i = 1, 2. The index set I(S, S1, S2)

in this case can be partitioned into the following subsets:

• I(S, S1, S2)0 = {(k1, l1, k2, l2) ∈ I(S, S1, S2) : (ki, li) 6= ∅ for some i =

1, 2};• I(S, S1, S2)1 = {(∅, ∅)}.

As before, ∅ is used as a symbol replacing a pair in these defintions.

For projective subspaces P1 ⊆ P2 ⊆ Pm+1,c we introduce the following notation.

(P2 \ P1) a Vk,l,c if and only if (P2 \ P1) ∩ Vk,l,c 6= ∅ and k is maximal with this

property. This is the smallest cell whose intersection with P2 is larger than that with

P1.

Recall the truncated Young wall Y and the morphism T : ZY → ZY from 6.3.

The following statement will be proved below in 7.4.

Theorem 7.2. There is a decomposition into locally closed subspaces

ZY =⊔

(k1,l1,k2,l2)∈I(S,S1,S2)

ZY (k1, l1, k2, l2),

where

ZY (k1, l1, k2, l2) = {I ∈ ZY : ((I∩Pm,j)\(I∩Pm,j))∩Pm+1,ci a Vki,li,ci for i = 1, 2}.

The symbol ∅ = (ki, li) means that there is no intersection with Pm+1,ci. Moreover, if

(k1, l1, k2, l2) ∈ I(S, S1, S2)e, then the nonempty fibers of T : ZY (k1, l1, k2, l2) → ZYhave Euler characterestic e.

The space ZY (k1, l1, k2, l2) should be thought as the space of those ideals, where

the generator in the cell of support block at position (0,m) has an image on the

(m + 1)-st diagonal at the cells Vki,li,ci for i = 1, 2 which does not come from the

rows above. We remark that the mentioned support block in the bottom row is also

salient since it is the first missing block in its row. But if one attaches further rows

to the bottom of Y , then it may become non-salient.

Recall from Section 7.1 the set Sci = Sb,0 ∪ Sb,ci ∪ Sb,c1∪c2 of support blocks for

label ci. In the light of the definition of the sets I(S, S1, S2) and Theorem 7.2 we

will say that the blocks in Sci can support the salient blocks of label ci in Ssi in the

following sense.


Each support block b ∈ S for label ci can support at most one salient block of

label ci above and to the left of b on the (m+ 1)-st antidiagonal. More precisely, this

supporting relationship has to respect the following supporting rules.

• Each block in Sb,0 supports precisely one or two salient blocks, at most one

from each label, and at least one of these has to be immediately above it;

• each block in Sb,ci can support at most one salient block of label ci which is

not immediately above it;

• each block in Sb,c1∪c2 can support none, one or two salient blocks, at most

one from each label, and neither of these is immediately above it.

In this way we define a correspondence from a subset of Sc1 to Ss1 and one from a

subset of Sc2 to Ss2 but these two have to satisfy the restrictions mentioned above

on the intersection Sc1 ∩ Sc2 = Sb,0 ∪ Sb,c1∪c2 . Neither correspondence has to be

surjective or be defined on the whole domain, but, where they are defined, they

should be injective.

We shall call a salient block b′ ∈ Si of label ci supported, if the number of support

blocks for label ci in S which are below b is at least as much as the total number

of salient blocks of label ci in Si counted from the top left, including b′ itself. A

supported salient block b′ satisfying condition (2) above, so that there is a support

block in S immediately below b′, will be called directly supported. The others will be

called non-directly supported. The supporting relationship will be globalized for the

whole diagram in the notion of closing datum, to be defined in 8.2 below.

Recall that during the inductive process in the proof of Theorem 6.1, at each step

a new generator appears in the cell corresponding to the salient block in the bottom

row. Assume that for I ∈ ZY , (I ∩ Pm,j) ∩ Pm+1,ci = I ∩ Pm+1,ci for i = 1, 2. In this

case we will say that there is no generator of label ci on the (m+ 1)-st antidiagonal.

Let S, S1 and S2 be the index sets for V0,m,j, V1,m,c1 and V1,m,c2 respectively. Then

using inductively Theorem 7.2 for each block b ∈ Sci we get that there is at most

one block bi ∈ Ssi such that when the row of b is added to the Young wall, the

new generator in the cell of b has nontrivial image in the cell of bi. Conversely, for

each block bi ∈ Ssi there corresponds a support block b ∈ Sci determined by I. In

particular, this implies

Corollary 7.3. Assume that for I ∈ ZY , (I ∩ Pm,j) ∩ Pm+1,ci = I ∩ Pm+1,ci

for some i = 1, 2. Let S, S1 and S2 be the index sets for V0,m,j, V1,m,c1 and V1,m,c2

respectively.

(1) |Su,1i | ≤ |Sb,ci |+ |Sb,c1∪c2|;(2) every salient block of label ci is supported;

(3) to each salient block of label ci there corresponds a unique support block for

label ci in the way described above.

7.3. SPECIAL LOCI IN GRASSMANNIANS 81

7.3. Special loci in Grassmannians

We prepare the ground for the proof of Theorem 7.2 by analyzing the incidence

varieties of 6.2 in the case m ≡ 1 mod n− 2. Once again, we use the notations of

Proposition 6.12. The composition with the projection from VS ×VS1,c1 ×VS2,c2 to its

first factor, followed by the affine linear fibration ω : VS → VS, defines a projection

map pVS : VS × VS1,c1 × VS2,c2 → VS.

For i = 1, 2 let Si(U) = {(ki, li) ∈ Bm+1,ci : (U) ∩ Vki,li,ci = ∅} be the blocks

in the partial profile of (U) ∩ Pm+1,ci on the (m + 1)-st diagonal. Then the index

sets I(S, S1(U), S2(U)) and I(S, S1(U), S2(U))e introduced above make sense. The

following lemma stratifies the fibers of the affine linear fibration ω : VS → VS.

Lemma 7.4. For any U ∈ VS, there is a stratification

V0,m,j/U =⊔

(k1,l1,k2,l2)∈I(S,S1(U),S2(U))

V0,m,j(k1, l1, k2, l2)/U,

where

V0,m,j(k1, l1, k2, l2)/U = {U ∈ V0,m,j/U : ((U)\(U))∩Pm+1,ci a Vki,li,ci for i = 1, 2}.

Moreover, if (k1, l1, k2, l2) ∈ I(S, S1(U), S2(U))e, then the space V0,m,j(k1, l1, k2, l2)/U

is of Euler characterestic e.

Proof. We have to distinguish the cases when S is maximal or not. The latter

case is significantly simpler, so we start with that.

If U ∈ VS with S not maximal, then (Mc1 + U) ∩ (Mc2 + U) = ∅ since Mc1 and

Mc2 are distinct parallel hyperplanes in V0,m,j, and there are affine subspaces Uirepresenting U such that Ui ⊆Mci . Recall from Corollary 6.29 the stratification of

V0,m,j which basically comes from the join structure on its closure Pm,j . This induces a

decomposition of V0,m,j/U into non-empty, locally closed, but not necessarily disjoint

spaces

V0,m,j/U =((V0,m,j \ (Mc1 ∩Mc2))/U) ∪ (Mc1/U) ∪ (Mc1/U)

⋃ ⋃(k1,l1)∈Bm+1,c1

((J(V0,m−1,c1 , Vk1,l1,c2) ∩ V0,m,j) \M0c1

+ U)/U

⋃ ⋃

(k2,l2)∈Bm+1,c1

((J(V0,m−1,c2 , Vk2,l2,c1) ∩ V0,m,j) \M0c2

+ U)/U

.

Consider a block (ki, li) ∈ Bm+1,ci \ Si(U). Then the intersection (U) ∩ Vki,li,ci 6= ∅.Assume that there is an U ∈ V0,m,j/U such that ((U) \ (U)) ∩ Vki,li,ci 6= ∅. Then

dim((U) ∩ Vki,li,ci) > dim((U) ∩ Vki,li,ci) so by Lemma 5.6 there is at least one other

block in a row above ki which has a trivial intersection with (U) but a nontrivial


one with (U). Hence, for any (ki, li) ∈ Bm+1,ci \ Si(U) we have

{U ∈ V0,m,j/U : ((U) \ (U)) ∩ Pm+1,ci a Vki,li,ci} = ∅.

On the other hand, if (ki, li) ∈ Si(U) ∪ {∅} then

((J(V0,m−1,ci , Vki,li,c3−i) ∩ V0,m,j) \M0ci

+ U)/U =

{U ∈ V0,m,j/U : ((U) \ (U)) ∩ Pm+1,ci a Vki,li,ci and ((U) \ (U)) ∩ Pm+1,c3−i a V1,m,c3−i}.

By dimension constrains these spaces are disjoint and it is easy to see that together

with (V0,m,j \ (Mc1 ∪Mc2))/U ,M0c1/U , and M0

c2/U they cover V0,m,j/U . Thus we get

a stratification

V0,m,j/U =V0,m,j(1,m, 1,m)/U

⊔ ⊔(k1,l1)∈(S1(U)∪{∅})\{(1,m)}

V0,m,j(k1, l1, 1,m)/U

⊔ ⊔

(k2,l2)∈(S2(U)∪{∅})\{(1,m)}

V0,m,j(1,m, k2, l2)/U

.

In particular, there is a stratification

(7.1) Mc3−i/U =⊔

(ki,li)∈(Si(U)∪{∅})\{(1,m)}

V0,m,j(ki, li, 1,m)/U.

Being an affine space, M0ci/U has Euler characteristic 1 for i = 1, 2. By Lemma 6.22

the spaces (J(V0,k(n−2),ci , Vki,li,c3−i) ∩ V0,m,j) \M0ci

have Euler characteristic 0, and

the same is true for ((J(V0,k(n−2),ci , Vki,li,c3−i) ∩ V0,m,j) \M0ci

+ U)/U . This last step

follows from the fact that the subspace U ⊂ Pm,j avoids both the image of Vki,li,c3−iand M0

ci.

If U ∈ VS such that S is maximal, then (Mc1 + U) = (Mc2 + U) = V0,m,j

since U is transversal to Mc1 and Mc2 . Therefore, there are two stratifications for

V0,m,j/U with i = 1, 2 as in (7.1). The claimed stratification is the largest common

refinement of these two. In particular, there are three types of strata. First, if

U ∈ ((J(V0,k(n−2),c1 , Vk1,l1,c2)∩ V0,m,j) \M0c1

+U)∩ ((J(V0,k(n−2),c2 , Vk2,l2,c1)∩ V0,m,j) \M0

c2+ U)/U for arbitrary (k1, l1) ∈ S1(U) and (k2, l2) ∈ S2(U), then ((U) \ (U)) ∩

Pm+1,ci a Vki,li,ci for i = 1, 2. Second, if U ∈ (((J(V0,k(n−2),c1 , Vki,li,c3−i) ∩ V0,m,j) +

U) \⋃

(k3−i,l3−i)∈S3−i(U)(J(V0,k(n−2),c2 , Vk3−i,l3−i,ci)∩ V0,m,j) +U)/U , then ((U) \ (U))∩Pm+1,ci a Vki,li,ci but ((U) \ (U)) ∩ Pm+1,c3−i = ∅. Third, if U ∈ ((M0

c1+ U) ∩ (M0

c2+

U))/U , then ((U)\(U))∩Pm+1,ci = ∅ for i = 1, 2. To sum it up, we get a stratification

into locally closed spaces

V0,m,j/U =⊔

(k1,l1)∈S1(U)∪{∅}(k2,l2)∈S2(U)∪{∅}

V0,m,j(k1, l1, k2, l2)/U.


The Euler characteristic of the stratum V0,m,j(∅, ∅)/U = ((Mc1 + U) ∩ (Mc2 + U))/U

is 1. It is left to the reader that the others have Euler characteristic 0. �

Let IS = {(S1(U), S2(U)) : U ∈ VS}. Actually IS only depends on S. For each

(S ′1, S′2) ∈ IS, let

VS(S ′1, S′2) = {U ∈ VS : (S1(U), S2(U)) = (S ′1, S

′2)}.

Corollary 7.5. For a fixed (S ′1, S′2) ∈ I(S) and (k1, l1, k2, l2) ∈ I(S, S ′1, S

′2) the

spaces V0,m,j(k1, l1, k2, l2)/U are isomorphic for every U ∈ VS(S ′1, S′2). Moreover, they

fit together into a locally closed subvariety VS(k1, l1, k2, l2) ⊆ VS which is a trivial

family over VS(S ′1, S′2). Using induction and the fact that the fiber product of locally

closed spaces is locally closed, we get that there is a stratification

VS =⊔

(S′1,S′2)∈IS

⊔(k1,l1,k2,l2)∈I(S,S′1,S′2)

VS(k1, l1, k2, l2)

into locally closed subvarieties. Furthermore, if (k1, l1, k2, l2) ∈ I(S, S ′1, S

′2)e, then the

fiber of ω : VS(k1, l1, k2, l2)→ VS has Euler characteristic e.

Proof. The triviality of the family VS(k1, l1, k2, l2) → VS(S ′1, S′2) follows from

Lemma 6.23 and the fact that V0,m,j(k1, l1, k2, l2)/U is constructed using (union,

intersection and difference of) joins in Pm,j . The rest of the statement is obvious. �

7.4. Proof of Theorem 7.2

As before, we fix S, S1 and S2. Recall that the fiber of the morphism ω × Id×Id : XS1,S2

S → Y S1,S2

Sover an element (U,U1, U2) ∈ Y S1,S2

Sis J(L−1

1 U1, L−11 U2)/U .

For i = 1, 2 let Si(U) = {(ki, li) ∈ Si : (U) ∩ Vki,li,ci = ∅} be the blocks in

the partial profile of (U) ∩ Pm+1,ci on the (m+ 1)-st diagonal. Then the index sets

I(S, S1(U), S2(U)) and I(S, S1(U), S2(U))e are defined. The following lemma, whose

proof is the same as that of Lemma 7.4, stratifies the fibers of the affine linear

fibration ω × Id× Id : XS1,S2

S → Y S1,S2

S.

Lemma 7.6. For any U1 ∈ VS1,c1 , U2 ∈ VS2,c2 the stratification of Lemma 7.4

restricts to a stratification

J(L−11 U1, L

−11 U2)/U =

⊔(k1,l1,k2,l2)∈I(S,S1(U),S2(U))

J(L−11 U1, L

−11 U2)(k1, l1, k2, l2)/U,

where

J(L−11 U1, L

−11 U2)(k1, l1, k2, l2)/U =

{U ∈ J(L−11 U1, L

−11 U2)/U : ((U) \ (U)) ∩ Pm+1,ci a Vki,li,ci for i = 1, 2}.

Moreover, if (k1, l1, k2, l2) ∈ I(S, S1(U), S2(U))e, then the space

J(L−11 U1, L

−11 U2)(k1, l1, k2, l2)/U is of Euler characterestic e.


Let IS1,S2

S = {(S1(U), S2(U)) : (U,U1, U2) ∈ Y S1,S2

S}. Actually IS1,S2

S only

depends on S, S1 and S2. For each (S ′1, S′2) ∈ IS1,S2

S , let

Y S1,S2

S(S ′1, S

′2) = {(U,U1, U2) ∈ Y S1,S2

S: (S1(U), S2(U)) = (S ′1, S

′2)}.

Corollary 7.7. For fixed (k1, l1, k2, l2) ∈ I(S, S ′1, S′2) the spaces

J(L−11 U1, L

−11 U2)(k1, l1, k2, l2)/U are isomorphic for every U ∈ Y S1,S2

S(S ′1, S

′2). More-

over, they fit together into a locally closed subvariety XS1,S2

S (k1, l1, k2, l2) ⊆ XS1,S2

S .

Using induction and the fact that the fiber product of locally closed spaces is locally

closed, we get that there is a stratification

XS1,S2

S =⊔

(S′1,S′2)∈IS1,S2

S

⊔(k1,l1,k2,l2)∈I(S,S′1,S′2)

XS1,S2

S (k1, l1, k2, l2)

into a locally closed subvarieties. Furthermore, if (k1, l1, k2, l2) ∈ I(S, S ′1, S

′2)e, then

the fiber of ω× Id× Id : XS1,S2

S (k1, l1, k2, l2)→ Y S1,S2

S(S ′1, S

′2) has Euler characteristic

e.

Proof of Theorem 7.2. With all these preparations the proof itself is very

easy. We just observe that ZY (k1, l1, k2, l2) consist of those points in ZY , which map

in (6.1) to XS1,S2

S (k1, l1, k2, l2) for some (S ′1, S′2) ∈ I

S1,S2

S such that (k1, l1, k2, l2) ∈I(S, S ′1, S

′2) . The result then follows from Corollary 7.7. �

CHAPTER 8

Type Dn: decomposition of the coarse Hilbert scheme

In this chapter we describe some distinguished subsets of the set of Young walls

Z∆ of type Dn. They will consist of Young walls which are the analogues of the

0-generated partitions from 4.2. Then we stratify the coarse Hilbert scheme. As

always, in the type D case there are substantial extra complications. Hence, we first

give a short guide to the chapter.

8.1. Guide to Chapter 8

Since the current chapter is rather technical, and it contains several definitions

and constructions, we present first a short guide explaining the motivation of the

definitions and also the main steps of the results. For the precise statements and

proofs see the body of the chapter in Sections 8.2-8.5. The reader is advised to

return to this guide during or after reading these later parts as well. The lemmas

and theorems are recollected with the same numbering as in the main body of the

chapter.

The injective set theoretical map i∗ : Hilb(C2/G∆)T → Hilb([C2/G∆])T =

tY ∈Z∆ZY induces a stratification

Hilb(C2/G∆)T = tY ∈Z∆WY ,

where WY := i−1∗ (ZY ∩ im(i∗)). Set WY = i∗(WY ).

We define the following subsets of Z∆.

(1) Z ′∆ contains those Young walls Y such WY 6= ∅, or equivalently, WY 6= ∅;(2) Z1

∆ these are called 0-generated Young walls (plays no role in Chapter 8);

(3) Z0∆ these are called distinguished 0-generated Young walls.

We have Z0∆ ⊂ Z1

∆ ⊂ Z ′∆. Neither of these is closed under the operation of bottom

row removal, which is the inductive step used in the proof of Theorem 6.3.

The characterization of WY or the computation of χ(WY ) is harder. It turns out

to group them into subsets indexed by Z0∆.

The main result of the chapter is

Theorem 8.14.

∞∑m=0

χ(Hilbm(C2/G∆)

)qm =

∑Y ∈Z0

∆

qwt0(Y ).

85

86 8. TYPE DN : DECOMPOSITION OF THE COARSE HILBERT SCHEME

Lemma 8.1. For each Y ∈ Z∆, there is a unique Y ′ ∈ Z ′∆ that contains Y and

is minimal with this property with respect to containment.

Lemma 8.2. There is a combinatorial reduction map red: Z ′∆ → Z0∆, restricting

to the identity on Z0∆ ⊂ Z ′∆, which associates to each Y ∈ Z ′∆ a distinguished

0-generated Young wall red(Y ) ∈ Z0∆ of the same 0-weight.

For a Young wall Y ∈ Z0∆, let Rel(Y ) = red−1(Y ) which is called the set of

relatives of Y .

In order to remedy the fact the sets Z0∆ and Z ′∆ are not closed under the operation

of bottom row removal we extend these families further and introduce the following

notions.

(1) A G∆-invariant ideal I is possibly invariant if it is generated by functions of

character ρ0,ρ1,ρn−1 and ρn “except for the bottom row”. The Young walls

of these are denoted as ZP∆.

(2) A G∆-invariant ideal I is almost invariant if it is in C[x, y]G∆ “except for the

bottom row”. The Young walls of these are denoted as ZA∆. In particular,

Z ′∆ ⊂ ZA∆ ⊂ ZP∆.

(3) Z0,A∆ ⊂ ZA∆ is a special subset of Young walls, which satisfy conditions

similar to that of Z0∆. In particular, Z0

∆ ⊂ Z0,A∆ .

There is analog of Lemma 8.2 for ZP∆ and Z0,A∆ , including a reduction ZP∆ → Z

0,A∆

and the notion of relatives. Let Rel(Y ) = red−1(Y ). Furthermore, the main advantage

of these new sets is the following.

Lemma 8.7. The sets ZP∆ and Z0,A∆ are closed under the operation of bottom row

removal.

One has the following commutative diagram:

Z0∆ ↪→ Z ′∆

↪→ ↪→

Z0,A∆ ↪→ ZA∆ ↪→ ZP∆.

The relatives of a Young wall Y ∈ Z0∆ are the same in Z ′∆ and in ZA∆, but there may

be some new relatives in ZP∆ which are not in ZA∆. This will cause no problem.

For each Y ∈ ZP∆, there is a locally closed decomposition:

ZY =⊔

d∈pcd(Y )

ZY (d)

Here pcd(Y ) is the set of all partial closing data defined on the support blocks of

Y . These are obtained with the inductive usage of Theorem 7.2 for each possibly

invariant ideal.

8.1. GUIDE TO Chapter 8 87

Accordingly, for each Y ∈ ZA∆ the almost invariant ideals are in a locally closed

subset WY ⊂ ZY and there is a locally closed decomposition

WY = td∈cd(Y )WY (d).

The set cd(Y ) of closing data are special partial closing data for Young walls in ZA∆,

in which all salient blocks of label 1, n− 1 or n are closed, except possibly one on

the bottom row.

The main ingredient for the proof of Theorem 8.14 is

Proposition 8.12. For all Y ∈ Z0,A∆ ,∑

Y ′∈Rel(Y )

χ(WY ′) = 1.

This implies that∑

Y ′∈Rel(Y ) χ(WY ′) = 1 for Y ∈ Z0∆, and can be rewritten as∑

Y ′∈Rel(Y )

∑d′∈cd(Y ′)

χ(WY ′(d′)) = 1.

The statement is proved in several steps.

Fix a Young wall Y ∈ ZP∆, and a partial closing datum d ∈ pcd(Y ). We say that

a support block for label c is of type e ∈ {−1, 0, 1} if, when its row is considered as

the bottom row, the associated half blocks according to d are in the set I(S1, S2, S)ein the notation of Theorem 7.2. The total number of support blocks of type e is

denoted as se(d).

Lemma 8.9. If the salient block b at the bottom row of Y is of type e ∈ {−1, 0, 1},then

χ(ZY (d)) = e · χ(ZY (d)).

Proposition 8.10. If Y ∈ ZP∆ and d ∈ pcd(Y ), then (with the notation 00 = 1)

χ(ZY (d)) = (−1)s−1(d) · 0s0(d) · 1s1(d).

Corollary 8.11. (1) Let Y ∈ ZP∆ and d ∈ pcd(Y ). If Y contains a

salient block of any label to which a support block not immediately below it

is associated under d, then χ(ZY (d)) = 0.

(2) Let Y ∈ ZA∆ and d ∈ cd(Y ).

(a) If Y contains a nondirectly supported salient block of label 1, n− 1 or n,

then χ(WY (d)) = 0.

(b) If Y does not contain any nondirectly supported salient block of label 1,

n− 1 or n, but d is not contributing, then χ(WY (d)) = 0.

By Corollary 8.11 (2.a), the strata associated to those Young walls Y ′ ∈ Rel(Y )

that have at least one undirectly supported salient block of label n−1 or n above the

bottom row do not contribute to the sum. Therefore we can restrict our attention to

the subset of Young walls in which the salient blocks not in the bottom row are


• directly supported salient block-pairs of label 0/1 or n− 1/n, or

• arbitrary salient blocks of label 0.

In particular, we can assume that all Young walls we are working with satisfy

(R1’): the salient blocks of label n− 1 or n not in the bottom row are all directly

supported. Moreover, by Corollary 8.11 (2.b) we can assume that each closing datum

is contributing. That is, for each salient block of label 1, n−1 or n not in the bottom

row, the associated support block is immediately below it, and this holds also for

those label 0 salient blocks which have an associated support block.

Then, we perform a case-by-case consideration.

8.2. Distinguished 0-generated Young walls

For a Young wall Y ∈ Z∆, denote by wt0(Y ) the 0-weight of Y , the number of

half-blocks labelled 0 in Y .

Recall from 6.3, respectively 7.1 the notions of a salient block and a support block

for a given label c ∈ {0, 1, n− 1, n}; we will use also all other notation introduced

in the latter section. We call a pair of salient half-blocks (b, b′) sharing the same

position a salient block-pair.

Consider the following conditions for a Young wall Y ∈ Z∆.

(A1) All salient blocks of Y are labelled 0, 1, n− 1 or n.

(A2) Every salient block of Y labelled c ∈ {1, n− 1, n} is supported.

Let Z ′∆ ⊂ Z∆ be the set of Young walls Y which satisfy conditions (A1)-(A2). We will

prove in Theorem 8.4, that Z ′∆ is the set of Young walls for which ZY ∩ im(i∗) 6= ∅.

Lemma 8.1. For each Y ∈ Z∆, there is a unique Y ′ ∈ Z ′∆ that contains Y and

is minimal with this property with respect to containment.

We will prove this statement at the end of the section.

Given a Young wall Y ∈ Z∆, a closing datum for Y is a function d from the set

of the salient blocks of Y of label c ∈ {1, n− 1, n}, and some subset of the salient

blocks of Y with label 0, to the set of support blocks of Y , such that

• for each salient block b of label c for which d is defined, the associated

support block d(b) is a support block for label c, and lies on the previous

antidiagonal and in a lower row than that of b;

• for each fixed c ∈ {0, 1, n− 1, n} the different salient blocks of label c are

mapped to different support blocks;

• each support block for label c can support at most one salient block of label

c according to the supporting rules spelled out at the end of 7.1.

By condition (A2), for every Y ∈ Z ′∆ with a nonempty set of salient blocks the set

cd(Y) of closing data for Y is nonempty. If all salient blocks of Y of label 1, n− 1

or n are directly supported, then a closing datum d ∈ cd(Y) is called contributing, if

8.2. DISTINGUISHED 0-GENERATED YOUNG WALLS 89

to every salient block of label on which d is defined, it associates the support block

immediately below it.

We define two subsets of Z ′∆. Consider the following conditions for a Young wall

Y ∈ Z∆.

(R1) The salient blocks of Y of label n−1 or n are all part of a directly supported

salient block-pair.

(R2) Y has no salient block with label in the set {1, . . . , n− 2}.(R3) Any consecutive series of rows of Y having equal length and ending in half

n − 1/n-blocks is longer than n − 2, or n − 1 if the length of the rows is

n − 1, and the last one starts with a block labelled 1 (see Example 8.15

below for the latter condition being broken).

Young walls satisfying (R1)–(R2) will be called 0-generated. Let Z1∆ ⊂ Z∆ denote

the set of 0-generated Young walls. They automatically satisfy (A1)–(A2), so indeed

Z1∆ ⊂ Z ′∆. Let further Z0

∆ ⊂ Z1∆ be the set of those Young walls which in addition

satisfy (R3) also. These will be called distinguished.

Lemma 8.2. There is a combinatorial reduction map red: Z ′∆ → Z0∆, restricting

to the identity on Z0∆ ⊂ Z ′∆, which associates to each Y ∈ Z ′∆ a distinguished

0-generated Young wall red(Y ) ∈ Z0∆ of the same 0-weight.

Proof. Starting with a Young wall Y ∈ Z ′∆, we construct red(Y ) by enforcing

(R1)–(R2) and (R3) in turn, making sure in the second step that (R1)–(R2) remain

fulfilled.

First, if a Young wall Y ∈ Z ′∆ violates (R1) or (R2) at a salient block of label

1, n− 1 or n, find the lowest row where this happens, and extend Y by adding as

many extra blocks as possible to this row without modifying the 0-weight. Thus, the

extension stops either just before any full block below which there is a missing full

block, or just before the next 0 block in the row, whichever comes earlier. Then in

the row above this, one or two blocks may become salient. If at least one of these

new salient blocks is not of label 0, then we repeat the same procedure. Following

this procedure all the way to the top of Y gives a new Young wall which satisfies

(R1) and (R2). These moves do not increase the number of places where the Young

wall violates (R3).

Second, assume a Young wall Y satisfies (R1) and (R2) but violates (R3): there

is a consecutive series of rows having equal length and ending in half n− 1/n-blocks,

but the length of this series is m ≤ n− 2. Remove the half block from end of the

lowest row of such a series. Then a new supported salient block-pair appears. If

the block b immediately above this block-pair is contained in Y , then we remove b,

as well as the blocks to the right of it in order to obtain a valid Young wall. Any

full block above b also cannot be present in a valid Young wall, so we remove that

also, as well as all blocks to the right. Continue the removal process until there is


a full block above the last removed block. This process terminates after m steps,

when it arrives at a row which is already short enough. In this way we decreased

the number places where the Young wall violates (R3), but haven’t increased the

number of places where it violates (R1) or (R2). The 0-weight of the Young wall

remains unchanged, since the length of the series was at most n− 2.

Combining these steps, we obtain a Young wall red(Y ) that satisfies (R3) as well

as (R1) and (R2), and so lies in Z0∆, and has the same 0-weight. �

Example 8.3. Let n = 4 and let us consider the following six Young walls.

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

0

1

0

1

0

1

43

43

43

34

34

01

01

01

10

10

43

43

34

4

3

4

1

Y1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

0

1

0

1

0

1

43

43

43

34

34

34

01

01

01

10

10

43

43

34

3

4

1

1

Y2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

0

1

0

1

0

1

43

43

43

34

34

01

01

01

10

10

43

43

34

3

4

3

1

Y3

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

0

1

0

1

0

1

43

43

43

34

34

34

01

01

01

10

10

43

43

34

4

3

1

1

Y4

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

0

1

0

1

0

1

43

43

43

34

34

34

01

01

01

10

10

43

43

34

1

Y5

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

0

1

0

1

0

1

43

43

43

34

34

34

01

01

01

10

10

43

43

34

1

1

Y6

It can be checked that all these satisfy conditions (A1) and (A2), and hence lie in

Z ′∆. On the other hand, the Young walls Y1 and Y3 violate (R1), and are extended to

Y2, resp. Y4 in the first step of the reduction process. Both of these still violate (R3);

in the second step, they become red(Y1) = red(Y3) = Y6 ∈ Z0∆. Y5 satisfies (R1), but

violates (R2) and is extended to Y6 in the first step with red(Y5) = Y6 ∈ Z0∆ also.

In each case the bullets indicate where exactly these violations occur. The blocks

without numbers are not contained in the Young walls.

8.3. THE DECOMPOSITION OF THE COARSE HILBERT SCHEME 91

For a Young wall Y ∈ Z0∆, let Rel(Y ) = red−1(Y ) denote the set of relatives of

Y , the Young walls we can get from Y by the inverses of the reduction steps above.

This is a finite directed set, directed by the steps of the proof of Lemma 8.2. In

Example 8.3, all Yi are relatives of Y6.

Proof of Lemma 8.1. Fix a Young wall Y ∈ Z∆. The positions of its 0 blocks

determine uniquely a Young wall Y1 ∈ Z0∆ such that wt0(Y ) = wt0(Y1), and the

0 blocks in Y1 are exactly the 0 blocks in Y . Y1 does not necessarily contain Y .

Consider the set RelY (Y1) ⊂ Z ′∆ of those relatives of Y1 which contain Y . This set

is nonempty, since we can always extend Y1 with the inverse of the move (R3) in

Lemma 8.2 until there are only label 0 salient blocks. There can be several of these

since there is an ambiguity in the inverse of the move (R3), but there is no Young

wall having the same 0 weight as Y1 which is not contained in at least one of these

extended Young walls.

Suppose that RelY (Y1) has two distinct minimal elements Y2, Y3 with respect

to containment. Then there is at least one row ending in a half block, where one

of Y2, Y3 has a left triangle, and the other has a right triangle, but otherwise the

row has the same length. Then the length of the series of successive rows with the

same length is the same in the two Young walls. If this length is more than n− 2,

then they cannot both contain Y . If it is n − 2 or less, then Y2, Y3 are the two

results of the inverse of the move (R3) applied on a smaller Young wall. Since Y

was a Young wall, also this smaller Young wall contains Y . Hence, neither of Y2, Y3

could be minimal. The same reasoning applies to all places where there is the left

triangle/right triangle ambiguity. Thus there is a unique minimal element in the set

of relatives of Y1 containing Y . �

8.3. The decomposition of the coarse Hilbert scheme

Let us turn to the Hilbert scheme of points on the quotient C2/G∆, the

coarse Hilbert scheme Hilb(C2/G∆) = tnHilbn(C2/G∆). Recall that the inclusion

C[x, y]G∆ ⊂ C[x, y] defines a morphism

p∗ : Hilb([C2/G∆])→ Hilb(C2/G∆), J 7→ JG∆ = J ∩ C[x, y]G∆

and a map of sets

i∗ : Hilb(C2/G∆)(C)→ Hilb([C2/G∆])(C), I 7→ C[x, y].I

between the coarse and the orbifold Hilbert schemes.

The purpose of this section is to prove the following result.

Theorem 8.4. The decomposition of the equivariant Hilbert scheme

Hilb([C2/G∆]) from Theorem 6.1 induces a locally closed decomposition

Hilb(C2/G∆) =⊔

Y ∈Z′∆

Hilb(C2/G∆)Y


of the coarse Hilbert scheme Hilb(C2/G∆) into strata indexed bijectively by the set Z ′∆of Young walls of type Dn satisfying (A1)-(A2) above. The stratum Hilb(C2/G∆)Yis contained in the m-th Hilbert scheme Hilbm(C2/G∆) for m = wt0(Y ).

Proof. We start with the universal ideal J �OHilb(C2/G∆) ⊗ C[x, y]G∆ , which

exists since Hilb(C2/G∆) is a fine moduli space. Using the relative pullback, we

obtain an invariant ideal C[x, y].J �OHilb(C2/G∆) ⊗ C[x, y], which however is not a

flat family of invariant ideals over Hilb(C2/G∆). Take the flattening stratification of

the base with index set F , to obtain a decomposition

(8.1) Hilb(C2/G∆) = tf∈FHilb(C2/G∆)f

over which the restrictions (C[x, y].J )f are flat. These flat families of invariant ideals

of C[x, y] define classifying maps

if : Hilb(C2/G∆)f → Hilbρ([C2/G∆])

from these strata to components of the equivariant Hilbert scheme. The latter smooth

varieties are decomposed into locally closed strata by Theorem 6.1 as

(8.2) Hilb([C2/G∆]) = tY ∈∈Z∆Hilb([C2/G∆])Y .

The stratification (8.2) gives a stratification on im(if ) for each f ∈ F since over each

Hilb([C2/G∆])Y the classifying map is flat. Hence, we can pull it back to obtain a

decomposition

Hilb(C2/G∆) =⊔

Y ∈Z∆

Hilb(C2/G∆)Y ,

where we have, set-theoretically,

Hilb(C2/G∆)Y (C) = {I ∈ Hilb(C2/G∆)(C) : i∗(I) ∈ Hilb([C2/G∆])Y (C)}.

The whole construction is compatible with the T = C∗-action, so we can also

decompose the T -fixed locus representing homogeneous ideals as

Hilb(C2/G∆)T =⊔

Y ∈Z∆

WY ,

where

WY (C) = {I ∈ Hilb(C2/G∆)(C) : i∗(I) ∈ ZY (C)}.Notice also that by construction, the maps iρ above are given by the pullback

map i∗. In other words, when restricted to the strata Hilb(C2/G∆)Y ⊃ WY , the

map i∗ becomes a morphism of schemes. On the other hand, it is also clear that,

letting WY denote the image of i∗ inside ZY , p∗ and i∗ are two-sided inverses and so

WY∼= WY ⊂ ZY .

To conclude, we need to show that for a fixed I ∈ Hilb(C2/G∆), the Young

wall Y associated with the pullback ideal J = i∗(I) necessarily lies in Z ′∆. It is

clearly enough to assume that I, and so J , are homogeneous. The ideal J , being

a pullback, is of course generated by invariant polynomials. On the other hand, as

8.4. POSSIBLY AND ALMOST INVARIANT IDEALS 93

we have seen during the proof of Theorem 6.1, a homogeneous ideal is generated by

polynomials lying in salient blocks. While not all salient blocks necessarily contain

a new generator, it is clear that salient blocks labelled j with 2 ≤ j ≤ n− 2 must

contain a generator. Since such a generator is not allowed in an invariant ideal, Y

must satisfy condition (A1).

To discuss the other condition (A2), let us return to the inductive proof of

Theorem 6.1. Corollary 7.3 (2) implies that if there is no generator on a given

antidiagonal of Y , then the salient blocks of label c ∈ {1, n, n−1} on this antidiagonal

are supported. For an invariant ideal, this condition is required to be satisfied for all

salient blocks of label c ∈ {1, n, n− 1}. This concludes the proof. �

8.4. Possibly and almost invariant ideals

We wish to study the Euler characteristics of the strata of the coarse Hilbert

scheme obtained in Theorem 8.4, using the inductive approach used in 6.3 in our

study of the orbifold Hilbert scheme. However, as things stand, the setup does not

lend itself well to induction based on the removal of the bottom row from a Young

wall, since the set of Young walls Z ′∆ is clearly not closed under the removal of the

bottom row. To remedy this, we introduce two auxiliary constructions. From now

on, except when noted, every ideal is supposed to be T -invariant.

First, call an ideal I � C[x, y] possibly invariant, if it is generated by

• polynomials which transform under G∆ according to ρ0, ρ1, ρn−1 or ρn,

• and at most one τ -invariant pair of polynomials of the same degree, forming

a two-dimensional representation of G∆, and not lying in the image of the

operator L1.

Equivalently, the second condition says that the corresponding two-dimensional

subspace lies in the large stratum of the appropriate projective space parameterizing

such subspaces.

Second, a possibly invariant ideal I will be called almost invariant, if it is

generated by

• G∆-invariant elements,

• and at most a single polynomial, or pair of polynomials of the same degree,

forming a one-or two-dimensional representation of G∆, and not lying in the

image of the operator L1.

Let us denote by ZP∆ ⊂ Z∆ the subset of all Young walls which are characterized

by the following condition:

(A1’) all salient blocks of Y are labelled 0, 1, n − 1 or n, except possibly for a

single salient block in the bottom row of a different label.

Moreover, let ZA∆ ⊂ ZP∆ be the set of Young walls which satisfy condition (A1’) as

well as the following second condition:


(A2’) every salient block of Y labelled c ∈ {1, n − 1, n} is supported, except

possibly the ones in the bottom row.

The following statement follows immediately from our setup.

Proposition 8.5.

(1) Possibly invariant ideals correspond to points in the strata ZY ⊂Hilb([C2/G∆]) where Y ∈ ZP∆.

(2) Points parameterizing almost invariant ideals lie in constructible subsets

WY ⊂ ZY of strata ZY ⊂ Hilb([C2/G∆]) for Y ∈ ZA∆.

By definition, we have Z ′∆ ⊂ ZA∆. For Y ∈ Z ′∆, the constructible subset provided

by Proposition 8.5(2) is exactly WY = i∗(WY ). Therefore, in the sequel we will

denote these strata as WY for arbitrary Y ∈ ZA∆ as well. Further, if for Y ∈ ZA∆, also

Y ∈ ZA∆, where Y is the Young wall obtained by removing from Y the bottom row

as in 6.3, then the map T : ZY → ZY of Lemma 6.16 takes WY to WY .

Furthermore, let Z0,A∆ ⊂ ZA∆ be the subset defined by the following conditions:

(R1’) the salient blocks of label n− 1 or n not in the bottom row are all directly

supported;

(R2’) there is no salient block with label in {1, . . . , n− 2} except possibly for the

bottom row;

(R3’) any consecutive series of rows, except the one starting in the bottom row,

having equal length and ending in half n− 1/n-blocks is longer than n− 2,

or n − 1 if the length of the rows is n − 1 and the last one starts with a

block labelled 1.

Lemma 8.6. There is a combinatorial reduction map red: ZP∆ → Z0,A∆ associating

to each Y ′ ∈ ZP∆ a unique Y ∈ Z0,A∆ .

Proof. The proof of Lemma 8.2 above goes through unchanged in this setting

and the reduction process gives a well-defined element in ZP∆. After the reduction

there is no indirectly supported salient block. Hence, each salient block is directly

supported, and in particular supported as required by condition (A2’). Therefore,

the output of the reduction is an element in ZA∆ with the properties (R1’)-(R3’). �

Once again, for Y ∈ Z0,A∆ the Young walls Rel(Y ) = red−1(Y ) will be called the

relatives of Y . The following statement is clear from the definitions.

Lemma 8.7. The sets ZP∆ and Z0,A∆ are closed under the operation of bottom row

removal.

8.4. POSSIBLY AND ALMOST INVARIANT IDEALS 95

The sets introduced so far can be placed in the following commutative diagram:

Z0∆ ↪→ Z ′∆

↪→ ↪→

Z0,A∆ ↪→ ZA∆ ↪→ ZP∆.

The relatives of a Young wall Y ∈ Z0∆ are the same in Z ′∆ and in ZA∆, but there

may be some new relatives in ZP∆. This will cause no problem. Even though we are

interested in strata of the Young walls in the upper row, it is easier to work in the

lower row because of Lemma 8.7.

The notion of a closing datum generalizes word by word for ideals in the stratum

of Y ∈ ZA∆. For ideals in the stratum of Young walls in ZP∆ we have to relax it,

since not all salient blocks of label 1, n − 1 or n are supported. A partial closing

datum for a Young wall in ZP∆ is given by associating to some of its salient blocks of

label c ∈ {0, 1, n− 1, n} a support block for label c in the previous antidiagonal and

below the salient block, in such a way that to each support block for label c at most

one salient block of label c is associated. We say that those salient blocks to which

there is an associated support block are closed. The set of all partial closing data

for Y ∈ ZP∆ will be denoted as pcd(Y ). Closing data are special partial closing data

for Young walls in ZA∆, in which all salient blocks of label 1, n− 1 or n are closed,

except possibly one on the bottom row.

Fix a Young wall Y ∈ ZP∆ such that in the bottom row the salient block is

a support block for label c ∈ {0, 1, n − 1, n}. Let Y be the truncation of Y . By

Lemma 8.7, Y ∈ ZP∆. If d is a partial closing datum associated to some I ∈ ZY , then

using the decomposition of Theorem 7.2 we can extend it to a partial closing datum

for each ideal in the fiber over I; in particular, if I ∈ ZY (k1, l1, k2, l2) for some pairs

(k1, l1) and (k2, l2), and either of these is a salient block of label c, then we associate

to them the support block in the bottom row. By induction, we obtain

Corollary 8.8. Given Y ∈ ZP∆, every I ∈ ZY defines a unique partial closing

datum d(I) ∈ pcd(Y ).

For d ∈ pcd(Y ), let ZY (d) ⊆ ZY be the subset of those ideals which have partial

closing datum d. Then

ZY = td∈pcd(Y )ZY (d)

is a locally closed decomposition of the stratum ZY . Similarly, for an element Y ∈ ZA∆let WY (d) ⊆ WY be the subset of those ideals which have closing datum d. Then

WY = td∈cd(Y )WY (d)

is a locally closed decomposition.


8.5. Euler characteristics of strata and the coarse generating series

In this section, we derive information about the topological Euler characteristics

of the strata of the coarse Hilbert scheme constructed above.

Fix a Young wall Y ∈ ZP∆, and a partial closing datum d ∈ pcd(Y ). We say that

a support block for label c is of type e ∈ {−1, 0, 1} if, when its row is considered as

the bottom row, the associated half blocks according to d are in the set I(S1, S2, S)ein the notation of Theorem 7.2.

Lemma 8.9. Assume that (Y, d) are such such that in the bottom row of Y , the

salient block b of label j ∈ {2, n− 2} is a support block for label c ∈ {0, 1, n− 1, n}.Let (Y , d) be the truncation of (Y, d). If the salient block b is of type e ∈ {−1, 0, 1},then

χ(ZY (d)) = e · χ(ZY (d)).

Proof. Using the notations of 7.2, let (k1, l1, k2, l2) be the quadruple of the half

blocks associated to the support block, and consider the diagram

ZY (d) ⊆ ZY −−→ XYyT yω×Id

ZY (d) ⊆ ZY −−→ YY .Returning once again to the process proving Theorem 6.1, when we obtained ZY from

the truncation ZY , we saw that those ideals in ZY that does not have a generator

in the strata of the missing half blocks at (k1, l1) and at (k2, l2) are necessarily in

ZY (k1, l1, k2, l2) and all ideals in ZY (k1, l1, k2, l2) have this property. Formally, a

point of ZY over ZY (d) is in ZY (d) if and only if

((I ∩ Pm,j) \ (I ∩ Pm,j)) ∩ Pm+1,ci a Vki,li,ci for i = 1, 2.

Under the operation T the space ZY (d) necessarily maps onto ZY (d). Hence, we get

that

ZY (d) = ZY (d)×ZY ZY (k1, l1, k2, l2).

By Theorem 7.2 the fibers of T on ZY (k1, l1, k2, l2) have Euler characteristic e. Thus

χ(ZY (d)) = e · χ(ZY (d)).

�

For e ∈ {−1, 0, 1} and c ∈ {0/1, n− 1/n} let se(d, c) be the number of support

blocks for label c of type e, and let se(d) = se(d, 0/1) + se(d, n − 1/n). Applying

Lemma 8.9 inductively, we get the following.

Proposition 8.10. For Y ∈ ZP∆ and d ∈ pcd(Y ),

χ(ZY (d)) = (−1)s−1(d) · 0s0(d) · 1s1(d),

where we adopt the convention 00 = 1.

8.5. EULER CHARACTERISTICS OF STRATA AND THE COARSE GENERATING SERIES 97

Corollary 8.11.

(1) Let Y ∈ ZP∆ and d ∈ pcd(Y ). If Y contains a salient block of any label to

which a support block not immediately below it is associated under d, then

χ(ZY (d)) = 0.

(2) Let Y ∈ ZA∆ and d ∈ cd(Y ).

(a) If Y contains a nondirectly supported salient block of label 1, n− 1 or n,

then χ(WY (d)) = 0.

(b) If Y does not contain any nondirectly supported salient block of label 1,

n− 1 or n, but d is not contributing, then χ(WY (d)) = 0.

Proof. (1) follows from Proposition 8.10, and both parts of (2) follows from

(1). �

The main ingredient for calculating the coarse generating series is the following

statement.

Proposition 8.12. For all Y ∈ Z0,A∆ ,∑

Y ′∈Rel(Y )

χ(WY ′) = 1.

Recall from Section 8.4 that the relatives of a Young wall Y ∈ Z0∆ are the same

in Z ′∆ and in ZA∆, but there may be some new relatives in ZP∆ which are not in ZA∆.

On the other hand, since WY = ∅ for Y ∈ ZP∆ \ ZA∆ Proposition 8.12 implies

Corollary 8.13. For all Y ∈ Z0∆,∑

Y ′∈Rel(Y )

χ(WY ′) = 1.

Proof of Proposition 8.12. By Corollary 8.11 (2.a), the strata associated

to those Young walls Y ′ ∈ Rel(Y ) that have at least one undirectly supported

salient block of label n− 1 or n above the bottom row do not contribute to the sum.

Therefore we can restrict our attention to the subset of Young walls in which the

salient blocks not in the bottom row are

• directly supported salient block-pairs of label 0/1 or n− 1/n, or

• arbitrary salient blocks of label 0.

In particular, we can assume that all Young walls we are working with satisfy

(R1’). Moreover, by Corollary 8.11 (2.b) we can assume that each closing datum is

contributing. That is, for each salient block of label 1, n− 1 or n not in the bottom

row, the associated support block is immediately below it, and this holds also for

those label 0 salient blocks which have an associated support block.

As in the proof of Theorem 6.3, we will prove the proposition by induction on

the number of rows. Fix a Young wall Y ∈ Z0,A∆ . We re-write the statement into the


form ∑Y ′∈Rel(Y )

∑d′∈cd(Y ′)

χ(WY ′(d′)) = 1.

Let Y be the truncation of Y ; by Lemma 8.7, Y ∈ Z0,A∆ also.

Assume first that the block above the salient block in the bottom row of Y is not

divided. Then the relatives of Y are exactly the extensions of those of Y with the

bottom row of Y , a closing datum on any such Young wall extends uniquely to the

extended Young wall, and the corresponding strata are isomorphic. In this case, the

induction step is obvious.

Assume next that the block above the salient block in the bottom row of Y is

divided. Let S be the index set of the block at (0,m), S1 and S2 the index sets for

the blocks at (1,m). The following cases can occur:

(1) S is maximal;

(2) S is not maximal, and exactly one of S1 or S2 is maximal;

(3) S is not maximal, and both S1 and S2 are maximal.

If S is maximal, then to each relative Y′ ∈ Rel(Y ) there corresponds a unique

relative Y ′ ∈ Rel(Y ) which satisfies (R1’) and (R2’): we extend each relative of

Y with the bottom row of Y . A closing datum d′

on one of these relatives Y′

can

be extended to a closing datum d′ on the corresponding relative of Y by assigning

(∅, ∅) to the new salient block appearing in the bottom row. By Theorem 7.2,

(∅, ∅) ∈ I(S, S1, S2)1, and∑Y ′∈Rel(Y )

∑d′∈cd(Y ′)

χ(WY ′(d′)) =

∑Y′∈Rel(Y )

∑d′∈cd(Y

′)

χ(WY′(d′)) = 1.

If S is not maximal, and Si is maximal while S3−i is not, then the missing block

at (1,m) is not salient, and the subspace in its stratum is necessarily in the image

of the subspace at (0,m). Again, to each relative of Y there corresponds a unique

relative of Y , and∑Y ′∈Rel(Y )

∑d′∈cd(Y ′)

χ(WY ′(d′)) =

∑Y′∈Rel(Y )

∑d′∈cd(Y

′)

χ(WY′(d′)) = 1.

Finally, suppose that S is not maximal, but S1 and S2 are maximal. In this case

new relatives appear when we add the bottom row of Y to Y . We investigate the

two possible cases individually.

If the divided block above the new salient block is a full n− 1/n block, then for

each relative of Y there are two other relatives, which contain either the label (n− 1)

half block or the label n half block. If the relatives of Y are {Y 0 = Y , Y 1, Y 2, . . . },then the relatives of Y are

{Y0 = Y, Y1, Y2, . . . } ∪⋃

c∈{n−1,n}

{Y0,c = Yc, Y1,c, Y2,c, . . . }.

8.5. EULER CHARACTERISTICS OF STRATA AND THE COARSE GENERATING SERIES 99

We obtain these by performing the inverse of the move (R3) in the algorithm of

Lemma 8.2. The addition of the bottom row of Y to an Yi schematically looks like

this:

Y i

YiYi,n−1 Yi,n

Here the block of label c is in the complement of the Young wall Yi,c, while its pair

is in it. There is only one contributing closing datum on each of Yi, Yi,n−1 and Yi,nextending a contributing closing datum di on Y i. Namely, to the new support block

of Yi we associate both the divided block above it, and every other part of di is kept

constant. We denote these by di, di,n−1 and di,n, respectively.

We claim that for c ∈ {n− 1, n},

χ(WYi,c(di,c)) = χ(WY i(di)).

To show this, we define a morphism WY i(di) → WY i,c

(di,c), where Y i,c is the

truncation of Yi,c. This morphism is given by the restriction of an ideal I ∈ WY i(di)

to the union of those cells whose block is missing from Y i,c. The Young wall Y i,c

has the same salient blocks as Y i except a half block in the bottom row. Hence, the

result is necessarily an ideal, which has the same generators as I except for the cell

V0,m,c where it does not have any generator. Therefore, the image of this morphism

is in WY i,c(di,c); here (0,m) is the position of the salient block-pair in the bottom

row of Y i, which is also the position of the salient half block of label κ(c) in the

bottom row of Y i,c.

Assume that there are two ideals I, I ′ ∈ WY i(di) which map to the same element

of WY i,c(di,c) under this morphism. Then they only differ in the function at V0,m,c, or

more precisely in the point of V0,m,c/U which represents the subspace in V0,m,c. Here

U = I ∩Pm,c = I ′ ∩Pm,c. Then any ideal I ′′ which is their affine linear combination,

i.e. whose corresponding subspace in V0,m,c is a linear combination of those of I and

I ′, is also an element of WY i(di), mapped to the same ideal as I and I ′ under the

morphism. In particular, the fibers of the morphism are affine spaces. Taking into

account that χ(WYi,c(di,c)) = χ(WY i,c(di,c)), this proves the claim.


Thus,

χ(WYi(di)) + χ(WYi,n−1(di,n−1)) + χ(WYi,n(di,n))

= −χ(WY i(di)) + χ(WY i

(di)) + χ(WY i(di)) = χ(WY i

(di)),

where in the first equality we used Lemma 8.9.

Second, if the divided block above the new salient block is a full 0/1 block,

then for each relative of Y there is a new relative which contains the label 1 half

block (and possibly some other blocks above it). Let us denote the relatives of Y as

{Y 0 = Y , Y 1, Y 2, . . . }. Then the relatives of Y are

{Y0 = Y, Y1, Y2, . . . } ∪ {Y0,1, Y1,1, Y2,1, . . . }.

We obtain these by performing the inverse of the move (R2) in the algorithm of

Lemma 8.2. Schematically:

Y i

Yi Yi,1

In this case Yi,0 cannot appear as a relative, since that would change the 0-weight.

Given a contributing closing datum di on Y i, shifting it appropriately gives a unique

contributing closing datum di,1 on Yi,1. On Yi, di induces two contributing closing

data:

• di is obtained by associating the support block in the bottom row to the

salient half-block pair above it;

• d′i is obtained by associating the support block in the bottom row to the

salient half block of label 1 above it only.

Again, using Lemma 8.9 we get

χ(WYi(di)) + χ(WYi(d′i)) + χ(WYi,1(di,1))

= −χ(WY i(di)) + χ(WY i

(di)) + χ(WY i(di)) = χ(WY i

(di)).

Summing over these, we obtain in all cases∑Y ′∈Rel(Y )

∑d′∈cd(Y ′)

χ(WY ′(d)) =∑

Y′∈Rel(Y )

∑d′∈cd(Y

′)

χ(WY′(d)),

which proves the induction step. �

8.5. EULER CHARACTERISTICS OF STRATA AND THE COARSE GENERATING SERIES101

Putting everything together, we obtain the following result, the analogue of

Corollary 4.3 in type D.

Theorem 8.14. Let ∆ be of type Dn. Then the generating function of Euler

characteristics of the coarse Hilbert schemes of points of the corresponding singular

surface C2/G∆ is given by the combinatorial generating series∞∑m=0

χ(Hilbm(C2/G∆)

)qm =

∑Y ∈Z0

∆

qwt0(Y ).

Proof. We use the decomposition of Theorem 8.4. For Y ∈ Z ′∆, we have

Hilb(C2/G∆)TY = WY and thus χ(WY ) = χ(Hilb(C2/G∆)Y ). Now use Corollary 8.13

to sum the Euler characteristics of the strata WY along the fibres of the combinatorial

reduction map red of Lemma 8.2. �

Example 8.15. Let n = 4 and let Y ∈ Z0∆ be the distinguished 0-generated

Young wall

2

2

2

2

2

2

0

1

0

1

0

43

43

34

1

The parameter space ZY of this Young wall Y is isomorphic to that of Example 6.6,

which in turn is isomorphic to that of Example 6.5. In particular, ZY ∼= A1. The

difference is that in this case the salient blocks are the 0-labelled blocks at (0, 4) and

(1, 5).

Denote by Y3 and Y4 the 0-generated non-distinguished Young walls

2

2

2

2

2

2

2

2

0

1

0

1

0

1

43

43

34

1

3

4

3

2

2

2

2

2

2

2

2

0

1

0

1

0

1

43

43

34

1

4

3

4

We have Y3, Y4 ∈ Z ′∆ and are in fact 0-generated, but both violate condition (R3)

at their fourth row, so they are not distinguished. In the first step of the reduction

algorithm of Lemma 8.2, we remove the half block of label 3 (resp., 4) from Y3 (resp,

Y4). Then we remove the blocks of label 2 and 4 (resp., 2 and 3) from the fifth row.

Finally we remove the blocks of 1,2 and 3 (reps., 1,2 and 4) from the sixth row. This

shows that both Young walls Y3, Y4 reduce to Y . It is not difficult to see that these are

in fact all the relatives of Y . As explained in Example 6.6, when the two generators


(v0, v′0) ∈ V0,4,0 × V1,5,0 are in a special position such that L1v0, v

′0 ∈ (L1L3), then

the ideal (v0, v′0) has Young wall Y3. Similarly, if L1v0, v

′0 ∈ (L1L4), then (v0, v

′0)

has Young wall Y4. In fact we can think of the corresponding strata as gluing

to one stratum inside the invariant Hilbert scheme Hilb3(C2/D4), the strata of Y3

and Y4 “patching in” the gaps of the stratum of Y . At least on the level of Euler

characteristics, this is what Proposition 8.12 shows in full generality.

CHAPTER 9

Type Dn: abacus combinatorics

In this chapter we carry out all the combinatorial calculations that are left from

the proofs of Theorem 3.2 and Theorem 3.5 in the type D case. Fortunately, an

abacus representation of the Young walls in type D has already been developed by

other authors [39, 41]. We carry out the enumerations in both the equivariant and

the coarse case on the abacus. Along the way, we confine the abacus representation

of 0-generated Young walls.

9.1. Guide to Chapter 9

Since the statements and methods in this chapter are again quite technical, we

start with a short summary or extended abstract. Again, the lemmas and theorems

are recollected with the same numbering as in the main body of the chapter.

Associated with each Young wall Y ∈ Z∆ there is an abacus configuration on an

abacus specific to the type D case. As in the type A case, there are certain sets of

blocks called bars on Y corresponding to the regular representation of G∆. These

bars can be removed from Y , and the removals correspond to operations on the

associated abacus. A Young wall Y ∈ Z∆ will be called a core Young wall, if no bar

can be removed from it without violating the Young wall rules. Let C∆ ⊂ Z∆ denote

the set of all core Young walls.

Proposition 9.2. Given a Young wall Y ∈ Z∆, any complete sequence of bar

removals through Young walls results in the same core core(Y ) ∈ C∆, defining a map

of sets

core : Z∆ → C∆.

The process can be described on the abacus, and results in a bijection

Z∆ ←→ Pn+1 × C∆,

where P is the set of ordinary partitions. Finally, there is also a bijection

(9.1) C∆ ←→ Zn.

Theorem 9.3. Let q = q0q1q22 . . . q

2n−2qn−1qn, corresponding to a single bar.

(1) The multi-weight qwt(Y ) =∏n

j=0 qwtj(Y )j of a core Young wall Y ∈ C∆ corre-

sponding to an element m = (m1, . . . ,mn) ∈ Zn under the correspondence

(9.1) can be expressed as

qm11 · · · · · qmnn (q1/2)m

>·C·m,

103

104 9. TYPE DN : ABACUS COMBINATORICS

where C is the Cartan matrix of type Dn.

(2) The multi-weight generating series

Z∆(q0, . . . , qn) =∑Y ∈Z∆

qwt(Y )

of Young walls for ∆ of type Dn can be written as

Z∆(q0, . . . , qn) =

∞∑m=(m1,...,mn)∈Zn

qm11 · · · · · qmnn (q1/2)m

>·C·m

∞∏m=1

(1− qm)n+1

.

The main result of the chapter is

Theorem 9.8. Let ∆ be of type Dn, and let ξ be a primitive (2n− 1)-st root of

unity. Then the generating series of the set Z0∆ of distinguished 0-generated Young

walls is given in terms of the generating function of all Young walls by the following

substitution: ∑Y ∈Z0

∆

qwt0(Y ) = Z∆(q0, . . . , qn)∣∣∣q0=ξ2q, q1=···=qn=ξ

.

For the proof of this, we first introduce a map

p : Z∆ → Z0∆,

and describe it combinatorially on the abacus.

Besides the abacus representation we introduce another representation of dis-

tinguished 0-generated Young walls in terms of sequences of integer pairs. Given

Y ∈ Z0∆, let ti denote the total number of beads in the i-th row of its abacus

representation, and li the number of beads in the rightmost position of the i-th row.

We obtain a sequence of pairs (ti, li)i∈Z+ , only finitely many of which do not equal

(0, 0).

Corollary 9.7. Given Y ∈ Z0∆, the associated sequence of pairs (ti, li)i∈Z+

satisfies the following conditions.

(1) For all i, 0 ≤ li ≤ ti.

(2) For all i, if ti > 0, then either li = ti is even, or li is odd.

(3) Either∑

i ti is even, or t1 − l1 = 2n− 4.

Conversely, any sequence (ti, li)i∈Z+ satisfying these conditions arises as a sequence

associated to at least one Young wall Y ∈ Z0∆. More precisely, the number of different

Young walls Y ∈ Z0∆ corresponding to any given sequence is 2m, where m is the

number of indices i such that ti − li > 2n− 2. All Young walls Y corresponding to a

single sequence have the same multi-weight, when the weights for labels n− 1 and n

are counted together.

9.2. YOUNG WALLS AND ABACUS OF TYPE DN 105

Corollary 9.11. Let Y ∈ Z0∆ be a distinguished 0-generated Young wall

described by the data {(ti, li)i}. Then∑Y ′∈p−1

∗ (Y )

qwt(Y ′)∣∣∣q1=···=qn=ξ, q0=ξ2q

= qwt0(Y )ξ∑j 6=0(wtj(Y )−dim ρj ·wt0(Y ))−

∑i c(ti,li),

where c(ti, li) is an integer depending only on ti and li.

Then we show by reducing every Y ∈ Z0∆ to a distinguished 0-generated core

that

ξ∑j 6=0(wtj(Y )−dim ρj ·wt0(Y ))−

∑i c(ti,li) = 1.

By Corollary 9.11, this implies that∑Y ′∈p−1

∗ (Y )

qwt(Y ′)∣∣∣q1=···=qn=ξ, q0=ξ2q

= qwt0(Y ),

which proves Theorem 9.8.

9.2. Young walls and abacus of type Dn

We continue to work with the root system ∆ of type Dn, and introduce some as-

sociated combinatorics. From now on, we return to the untransformed representation

of the type Dn Young wall pattern introduced in 5.2.

Recalling the Young wall rules (YW1)-(YW4), it is clear that every Y ∈ Z∆

can be decomposed as Y = Y1 t Y2, where Y1 ∈ Z∆ has full columns only, and

Y2 ∈ Z∆ has all its columns ending in a half-block. These conditions define two

subsets Zf∆,Zh∆ ⊂ Z∆. Because of the Young wall rules, the pair (Y1, Y2) uniquely

reconstructs Y , so we get a bijection

(9.2) Z∆ ←→ Zf∆ ×Zh∆.

Given a Young wall Y ∈ Z∆ of type Dn, let λk denote the number of blocks (full

or half blocks both contributing 1) in the k-th vertical column. By the rules of Young

walls, the resulting positive integers {λ1, . . . , λr} form a partition λ = λ(Y ) of weight

equal to the total weight |Y |, with the additional property that its parts λk are

distinct except when λk ≡ 0 mod (n− 1). Corresponding to the decomposition (9.2),

we get a decomposition λ(Y ) = µ(Y ) t ν(Y ). In µ(Y ), no part is congruent to 0

modulo (n− 1), and there are no repetitions; all parts in ν(Y ) are congruent to 0

modulo (n− 1) and repetitions are allowed. Note that the pair (µ(Y ), ν(Y )) does

almost, but not quite, encode Y , because of the ambiguity in the labels of half-blocks

on top of non-complete columns.

We now introduce the type1 Dn abacus, following [39, 41]. This abacus is the

arrangement of positive integers, called positions, in the following pattern.

1Once again, we should call it type D(1)n , but we simplify for ease of notation.


1 . . . n− 2 n− 1 n . . . 2n− 3 2n− 2

2n− 1 . . . 3n− 4 3n− 3 3n− 2 . . . 4n− 5 4n− 4...

......

......

...

For any integer 1 ≤ k ≤ 2n − 2, the set of positions in the k-th column of the

abacus is the k-th runner, denoted Rk. As in type A, positions on the runners are

occupied by beads. For k 6≡ 0 mod (n− 1), the runners Rk can only contain normal

(uncolored) beads, with each position occupied by at most one bead. On the runners

Rn−1 and R2n−2, the beads are colored white and black. An arbitrary number of

white or black beads can be put on each such position, but each position can only

contain beads of the same color.

Given a type Dn Young wall Y ∈ Z∆, let λ = µtν be the corresponding partition

with its decomposition. For each nonzero part νk of ν, set

nk = #{1 ≤ j ≤ l(µ) | µj < νk}

to be the number of full columns shorter than a given non-full column. The abacus

configuration of the Young wall Y is defined to be the set of beads placed at positions

λ1, . . . , λr. The beads at positions λk = µj are uncolored; the color of the bead at

position λk = νl corresponding to a column C of Y iswhite, if the block at the top of C is and nl is even;

white, if the block at the top of C is and nl is odd;

black, if the block at the top of C is and nl is even;

black, if the block at the top of C is and nl is odd.

One can check that the abacus rules are satisfied, that all abacus configurations

satisfying the above rules, with finitely many uncolored, black and white beads, can

arise, and that the Young wall Y is uniquely determined by its abacus configuration.

Example 9.1. The abacus configuration associated to the Young wall Y6 of

Example 8.3 is

R1 R2 R3 R4 R5 R6

1 2 3 4 5 67 8 9 10 11 1213 14 15 16 17 18

......

9.3. Core Young walls and their abacus representation

In parallel with the type A story, we now introduce the combinatorics of core

Young walls of type Dn, and the corresponding abacus moves. On the Young wall

side, define a bar to be a connected set of blocks and half-blocks, with each half-block

occuring once and each block occuring twice. A Young wall Y ∈ Z∆ will be called a

9.3. CORE YOUNG WALLS AND THEIR ABACUS REPRESENTATION 107

core Young wall, if no bar can be removed from it without violating the Young wall

rules. For an example of bar removal, see [39, Example 5.1(2)]. Let C∆ ⊂ Z∆ denote

the set of all core Young walls. The following statement is the type D analogue of

the discussion of 4.4.

Proposition 9.2. ([39, 41]) Given a Young wall Y ∈ Z∆, any complete sequence

of bar removals through Young walls results in the same core core(Y ) ∈ C∆, defining

a map of sets

core : Z∆ → C∆.

The process can be described on the abacus, respects the decomposition (9.2), and

results in a bijection

(9.3) Z∆ ←→ Pn+1 × C∆

where P is the set of ordinary partitions. Finally, there is also a bijection

(9.4) C∆ ←→ Zn.

Proof. Decompose Y into a pair of Young walls (Y1, Y2) as above. Let us first

consider Y1. On the corresponding runners Rk, k 6≡ 0 mod (n − 1), the following

steps correspond to bar removals [39, Lemma 5.2].

(B1) If b is a bead at position s > 2n − 2, and there is no bead at position

(s− 2n+ 2), then move b one position up and switch the color of the beads

at positions k with k ≡ 0 mod (n− 1), s− 2n+ 2 < k < s.

(B2) If b and b′ are beads at position s and 2n−2−s (1 ≤ s ≤ n−2) respectively,

then remove b and b′ and switch the color of the beads at positions k ≡0 mod (n− 1), s < k < 2n− 2− s.

Performing these steps as long as possible results in a configuration of beads on the

runners Rk with k 6≡ 0 mod (n− 1) with no gaps from above, and for 1 ≤ s ≤ n− 2,

beads on only one of Rs, R2n−2−s. This final configuration can be uniquely described

by an ordered set of integers {z1, . . . , zn−2}, zs being the number of beads on Rs minus

the number of beads on R2n−2−s [41, Remark 3.10(2)]. In the correspondence (9.4)

this gives Zn−2. It turns out that the reduction steps in this part of the algorithm

can be encoded by an (n− 2)-tuple of ordinary partitions, with the summed weight

of these partitions equal to the number of bars removed [39, Theorem 5.11(2)].

Let us turn to Y2, represented on the runners Rk, k ≡ 0 mod (n− 1). On these

runners, the following steps correspond to bar removals [41, Sections 3.2 and 3.3].

(B3) Let b be a bead at position s ≥ 2n − 2. If there is no bead at position

(s− n+ 1), and the beads at position (s− 2n+ 2) are of the same color as

b, then shift b up to position (s− 2n+ 2).

(B4) If b and b′ are beads at position s ≥ n− 1, then move them up to position

(s− n+ 1). If s− n+ 1 > 0 and this position already contains beads, then

b and b′ take that same color.


During these steps, there is a boundary condition: there is an imaginary position

0 in the rightmost column, which is considered to contain unvisible white beads;

placing a bead there means that this bead disappears from the abacus. It turns out

that the reduction steps in this part of the algorithm can be described by a triple of

ordinary partitions, again with the summed weight of these partitions equal to the

number of bars removed [41, Proposition 3.6]. On the other hand, the final result

can be encoded by a pair of 2-core partitions which appeared in Section 4.4.

The different bar removal steps (B1)-(B4) construct the map c algorithmically

and uniquely. The stated facts about parameterizing the steps prove the existence of

the bijection (9.3). To complete the proof of (9.4), we only need to remark further

that the set of 2-core partitions, in our language A1-core partitions, is in bijection

with the set of integers by bijection (4.11) in Section 4.4 (see also [41, Remark

3.10(2)]). This gives the remaining Z2 factor in the bijection (9.4). �

We next determine the multi-weight of a Young wall Y in terms of the bi-

jections (9.3)-(9.4). The quotient part is easy: the multi-weight of each bar is

(1, 1, 2, . . . , 2, 1, 1), so in complete analogy with the type A situation, the contribution

of the (n+ 1)-tuple of partitions to the multi-weight is easy to compute. Turning to

cores, under the bijection C∆ ↔ Zn, the total weight of a core Young wall Y ∈ C∆

corresponding to (z1, . . . , zn) ∈ Zn is calculated in [41, Remark 3.10]:

(9.5) |Y | = 1

2

n−2∑i=1

((2n− 2)z2

i − (2n− 2i− 2)zi)

+ (n− 1)n∑

i=n−1

(2z2

i + zi).

A refinement of this formula calculates the multi-weight of Y .

Theorem 9.3. Let q = q0q1q22 . . . q

2n−2qn−1qn, corresponding to a single bar.

(1) Composing the bijection (9.4) with an appropriate Z-change of coordinates

in the lattice Zn, the multi-weight of a core Young wall Y ∈ C∆ corresponding

to an element m = (m1, . . . ,mn) ∈ Zn can be expressed as

qm11 · · · · · qmnn (q1/2)m

>·C·m,

where C is the Cartan matrix of type Dn.

(2) The multi-weight generating series

Z∆(q0, . . . , qn) =∑Y ∈Z∆

qwt(Y )

of Young walls for ∆ of type Dn can be written as

Z∆(q0, . . . , qn) =

∞∑m=(m1,...,mn)∈Zn

qm11 · · · · · qmnn (q1/2)m

>·C·m

∞∏m=1

(1− qm)n+1

.


(3) The following identity is satisfied between the coordinates (m1, . . . ,mn) and

(z1, . . . , zn) on Zn:

n∑i=1

mi = −n−2∑i=1

(n− 1− i)zi − (n− 1)c(zn−1 + zn)− (n− 1)b.

Here z1 + · · ·+ zn−2 = 2a− b for integers a ∈ Z, b ∈ {0, 1}, and c = 2b− 1 ∈{−1, 1}.

The coordinate change (z1, . . . , zn) 7→ (m1, . . . ,mn) and the multiweight formula

of (1), as well as (3), follow from somewhat involved but routine calculations. (2)

clearly follows from (1) and the preceeding discussion.

Let us write zI =∑

i∈I zi for I ⊆ {1, . . . , n− 2}. Each such number decomposes

uniquely as zI = 2aI − bI , where aI ∈ Z and bI ∈ {0, 1}. Let us introduce also

cI = 2bI − 1 ∈ {−1, 1}. We will make use of the relations

aI =∑i∈I

ai −∑

i1∈I,i2∈I\{i1}

bi1bi2 +∑

i1∈I,i2∈I\{i1},i3∈I\{i1,i2}

2bi1bi2bi3 − . . . ,

bI =∑i∈I

bi −∑

i1∈I,i2∈I\{i1}

2bi1bi2 +∑

i1∈I,i2∈I\{i1},i3∈I\{i1,i2}

4bi1bi2bi3 − . . . .

To simplify notations let us introduce

rI := aI −∑i∈I

ai = −∑

i1∈I,i2∈I\{i1}

bi1bi2 +∑

i1∈I,i2∈I\{i1},i3∈I\{i1,i2}

2bi1bi2bi3 − . . . .

Using these notations the colored refinement of the weight formula (9.5) is the

following.

Lemma 9.4. Given a core Young wall Y ∈ C∆ corresponding to (zi) ∈ Zn in the

bijection of (9.4), its content is given by the formula

qwt(Y ) = q−∑n−2i=1 bi

1 q−2a1−

∑n−2i=2 bi

2 . . . q−∑n−3i=1 2ai−bn−2

n−2 (q0q−11 qn−1qn)−

∑n−2i=1 ai(q0q

−11 )a1...n−2

·q12

∑n−2i=1 (z2

i +bi)+z2n−1+z2

n

·(qb1...n−2(q−11 . . . q−1

n−2q−1n−1)c1...n−2)zn−1(qb1...n−2(q−1

1 . . . q−1n−2q

−1n )c1...n−2)zn .

Proof. When forgetting the coloring the lemma obviously gives back (9.5).

Notice also that z2i +bi = 4a2

i −4aibi+2bi is always an even number, so the exponents

are always integers. Later we will show that the expression is a multivariable theta

function.

The total weight of a core basically measures the area of its Young wall. So the

exponents can be at most quadratic in the parameters {zi}1≤i≤n. Hence, it is enough

to check that the formula is correct in two cases:

(1) when any of the zi’s is set to a given number and the others are fixed to 0;

and


(2) when all of the parameters are fixed to 0 except for an arbitrary pair zi and

zj, i 6= j.

First, consider that zi 6= 0 for a fixed i, and zj = 0 in case j 6= i.

(a) When 1 ≤ i ≤ n − 2, then the colored weight of the corresponding core

Young wall is

(q1 . . . qi)−bi(q2

i+1 . . . q2n−2qn−1qn)−aiq2a2

i−2aibi+bi .

(b) When i ∈ {n− 1, n}, then the associated core Young wall has colored weight

qz2i (q1q2 . . . qn−2qi)

zi .

Both of these follow from (9.5) and its proof in [41] by taking into account the colors

of the blocks in the pattern.

Second, assume that zi and zj are nonzero, but everything else is zero. Then the

total weight is not the product of the two individual weights, but some correction

term has to be introduced. The particular cases are:

(a) 1 ≤ i, j ≤ n− 2. There can only be a difference in the numbers of q0’s and

q1’s which comes from the fact that in the first row there are only half blocks

with 0’s in the odd columns and 1’s in the even columns. Exactly −rijblocks change color from 0 to 1 when both zi and zj are nonzero compared

to when one of them is zero. In general, this gives the correction term

(q0q−11 )r1...n−2 = (q0q

−11 )a1...n−2−

∑n−2i=1 ai .

(b) 1 ≤ i ≤ n− 2, j ∈ {n− 1, n}. For the same reason as in the previous case

the parity of zi modifies the colored weight of the contribution of zj , but not

the total weight of it. If zi is even, then the linear term of the contribution

of zj is q1q2 . . . qn−2qj . In the odd case it is q0q2 . . . qn−2qκ(j). This is encoded

in the correction term (qb1...n−2(q−11 . . . q−1

n−2q−1j )c1...n−2)zj .

(c) i = n− 1, j = n. zn−1 and zn count into the total colored weight completely

independently, so no correction term is needed.

Putting everything together gives the claimed formula for the colored weight of

an arbitrary core Young wall.

�

Now we turn to the proof of Theorem 9.3. After recollecting the terms in the

formula of Lemma 9.4 it becomes

q−b1...n−2−c1...n−2(zn−1+zn)1

n−2∏i=2

q−2a1...i−1+c1...i−1bi...n−2−c1...n−2(zn−1+zn)i

·q−a1...n−2−c1...n−2zn−1

n−1 q−a1...n−2−c1...n−2znn

·q∑n−2i=1 (2a2

i−2aibi+bi)+b1...n−2zn−1+z2n−1+b1...n−2zn+z2

n+r1...n−2


Let us define the following series of integers:

m1 = −b1...n−2 − c1...n−2(zn−1 + zn) ,

m2 = −2a1 + c1b2...n−2 − c1...n−2(zn−1 + zn) ,...

mn−2 = −2a1...n−3 + c1...n−3bn−2 − c1...n−2(zn−1 + zn) ,

mn−1 = −a1...n−2 − c1...n−2zn−1 ,

mn = −a1...n−2 − c1...n−2zn .

It is an easy and enlightening task to verify that the map

Zn → Zn, (z1, . . . , zn) 7→ (m1, . . . ,mn)

is a bijection, which is left to the reader.

Proof of Theorem 9.3. (1): One has to check that

n∑i=1

m2i −m1m2 −m2m3 − · · · −mn−2(mn−1 +mn) =

=n−2∑i=1

(2a2i − 2aibi + bi) + b1...n−2zn−1 + z2

n−1 + b1...n−2zn + z2n + r1...n−2 .

The terms containing zn−1 or zn on the left hand side are

(n− 2)(zn−1 + zn)2 + z2n−1 + z2

n − (n− 3)(zn−1 + zn)2 − z2n−1 − z2

n − 2zn−1zn

+

(2b1...n−2 +

n−3∑i=1

2(2a1...i − c1...ibi+1...n−2) + 2a1...n−2

)c1...n−2(zn−1 + zn)

−

(b1...n−2 +

n−3∑i=1

2(2a1...i − c1...ibi+1...n−2) + 2a1...n−2

)c1...n−2(zn−1 + zn)

= b1...n−2zn−1 + z2n−1 + b1...n−2zn + z2

n ,

since b1...n−2c1...n−2 = b1...n−2.

The terms containing neither zn−1 nor zn on the left hand side are

b1...n−2 +n−3∑i=1

(2a1...i − c1...ibi+1...n−2)2 + 2a21...n−2 − b1...n−2(2a1 − c1b2...n−2)

−n−4∑i=1

(2a1...i − c1...ibi+1...n−2)(2a1...i+1 − c1...i+1bi+2...n−2)

−2(2a1...n−3 − c1...n−3bn−2)a1...n−2 .

Lemma 9.5.

2a1...i − c1...ibi+1...n−2 =i∑

j=1

(2aj − bj) + b1...n−2 ,


Proof.

2a1...i − c1...ibi+1...n−2

= 2a1...i−1 + 2ai − 2b1...i−1bi + c1...i−1cibi+1...n−2

= 2a1...i−1 + 2ai − 2b1...i−1bi + 2c1...i−1bibi+1...n−2 − c1...i−1bi+1...n−2

= 2a1...i−1 + 2ai − bi − c1...i−1(bi+1...n−2 + bi − 2bibi+1...n−2)

= 2a1...i−1 − c1...i−1bi...n−2 + 2ai − bi ,

and then use induction. �

Applying Lemma 9.5 and the last intermediate expression in its proof to the

terms considered above, they simplify to

b1...n−2 +n−3∑i=1

(2a1...i − c1...ibi+1...n−2)2 + 2a21...n−2 − b1...n−2(2a1 − c1b2...n−2)

−n−4∑i=1

(2a1...i − c1...ibi+1...n−2)(2a1...i − c1...ibi+1...n−2 + 2ai+1 − bi+1)

−2(2a1...n−3 − c1...n−3bn−2)a1...n−2

= b1...n−2 + (2a1...n−3 − c1...n−3bn−2)2 + 2a21...n−2 − b1...n−2(2a1 − c1b2...n−2)

−n−4∑i=1

(2a1...i − c1...ibi+1...n−2)(2ai+1 − bi+1)− 2(2a1...n−3 − c1...n−3bn−2)a1...n−2

= b1...n−2 + (2a1...n−3 − c1...n−3bn−2)(2a1...n−3 − c1...n−3bn−2 − 2a1...n−2)

+2a21...n−2 − b1...n−2(2a1 − c1b2...n−2)−

n−4∑i=1

(2a1...i − c1...ibi+1...n−2)(2ai+1 − bi+1)

= b1...n−2 + 2a21...n−2 − b1...n−2(2a1 − c1b2...n−2)

−n−3∑i=1

(2a1...i − c1...ibi+1...n−2)(2ai+1 − bi+1)

= 2a21...n−2 − b1...n−2(2a1 − b1)−

n−3∑i=1

(i∑

j=1

(2aj − bj) + b1...n−2

)(2ai+1 − bi+1) .

Let us denote this expression temporarily as sn−2. Taking into account that

a1...n−2 = a1...n−3 + an−2 − b1...n−3bn−2 ,

b1...n−2 = b1...n−3 + bn−2 − 2b1...n−3bn−2 ,


sn−2 can be rewritten as

2a21...n−3 + 2a2

n−2 + 2b1...n−3bn−2 + 4a1...n−3an−2

−4a1...n−3b1...n−3bn−2 − 4an−2b1...n−3bn−2

−(b1...n−3 + bn−2 − 2b1...n−3bn−2)(2a1 − b1)

−n−4∑i=1

(i∑

j=1

(2aj − bj) + b1...n−3

)(2ai+1 − bi+1)

−n−3∑i=1

(bn−2 − 2b1...n−3bn−2)(2ai+1 − bi+1)

−

(n−3∑j=1

(2aj − bj) + b1...n−3

)(2an−2 − bn−2)

= sn−3 + 2a2n−2 + 2b1...n−3bn−2 + 4a1...n−3an−2

−4a1...n−3b1...n−3bn−2 − 4an−2b1...n−3bn−2

−(bn−2 − 2b1...n−3bn−2)

(n−2∑i=1

2ai − bi

)−

(n−3∑j=1

(2aj − bj) + b1...n−3

)(2an−2 − bn−2)

= sn−3 + 2a2n−2 − 2an−2bn−2 + bn−2 + 2an−2

(2a1...n−3 −

n−3∑j=1

(2aj − bj)

)+b1...n−3bn−2 − 2an−2b1...n−3

= sn−3 + 2a2n−2 − 2an−2bn−2 + bn−2 − b1...n−3bn−2 ,

where at the last equality the identity∑n−3

j=1 (2aj − bj) = z1...n−3 = 2a1...n−3 − b1...n−3

was used.

It can be checked, that s1 = 2a21 − 2a1b1 + b1, so induction shows that

sn−2 =n−2∑i=1

(2a2i − 2aibi + bi + b1...i−1bi) .

It remains to show thatn−2∑i=2

b1...i−1bi = r1...n−2 ,

which requires another induction argument, and is left to the reader.

(3): Apply Lemma 9.5 on

n∑i=1

mi = −b1...n−2−n−2∑i=1

(2a1...i−1−c1...i−1bi...n−2)−2a1...n−2−(n−1)c1...n−2(zn−1 +zn).

�


9.4. 0-generated Young walls and their abacus representations

In this section, we characterize the abacus configurations corresponding to Young

walls in the sets Z0∆ and Z1

∆ defined in 8.2.

Recall conditions (R1)–(R3) on Young walls Y ∈ Z∆ from 8.2. Recall also that a

Young wall corresponds uniquely to an abacus configuration, where the beads are

placed at the positions λ1, . . . , λr. Finally recall that the quantity nk denotes the

number of full columns shorter than a given non-full column of height λk.

Lemma 9.6. Conditions (R1)-(R2) on Young walls are equivalent to the following

conditions for an abacus configuration.

(D1) In each row, the rightmost bead is always on the (2n − 2)-nd ruler, and

either all the beads of the row are at this position, or the number of beads in

this position is odd.

(D2) If a row ends with a white (resp., black) bead corresponding to a column of

height λk, then k + nk must be odd (resp. even). If several beads are placed

on this position, which is allowed since it is on the ruler R2n−2, then this

condition refers to the smallest possible k.

(D3) The total number of beads in the whole abacus is even, or the total number

of beads on the rulers R1, . . . , Rn−1 in the first row is n− 2.

(D4) The beads on the rulers Rn . . . , R2n−3 are pushed to the right as much

as possible in each row. In any given row, the positions on the rulers

R1, . . . , Rn−1 are empty unless all the positions on the rulers Rn, . . . , R2n−2

are filled.

(D5) The beads on the rulers R1 . . . Rn−1 in any given row are either all on the

ruler Rn−1, or on the rules R1 . . . Rn−2, and pushed to the right as much as

possible.

Condition (R3) is equivalent to the following condition.

(D6) Let s be the total number of beads on the rulers R1, . . . , Rn−1 in any given

row.

(a) If s > n− 2, then all these beads are on Rn−1.

(b) If s ≤ n−2, then all these beads are on the rulers R1, . . . , Rn−2, pushed

to the right.

Thus 0-generated Young walls Y ∈ Z1∆ correspond to abacus configurations satisfying

(D1)-(D5), whereas distinguished 0-generated Young walls Y ∈ Z0∆ correspond to

those satisfying (D6) also.

Proof. Two kinds of salient blocks can appear in a Young wall that satisfies

(R1)-(R2):

• label 0 half-blocks,

• and salient block-pairs of label n− 1 or n.

9.4. 0-GENERATED YOUNG WALLS AND THEIR ABACUS REPRESENTATIONS 115

As in the type A case a salient block corresponds to the first bead in a consecutive

series of beads in the abacus. More precisely, if there are several columns of height

λk, or equivalently, if there are several beads placed on the position λk, then the

salient block corresponds to that column of height λk which has the smallest possible

index k among these.

The label 0 blocks always correspond to positions which are on the ruler R2n−2.

In the odd columns of the type D pattern they are of the shape while in the even

columns they are of the shape . Condition (D2) encodes the fact, that the salient

blocks of label 0 are upper triangles in odd columns and lower triangles in even

columns, and that the color of the beads corresponding to them is also affected by

the parity of the appropriate nk.

If there is a salient block of label 0, then some columns of the same height, let’s

say, λk, can follow it. If the first column after them has height λk − 1 then on its

top there is again a salient block of label 0. This block can only have the opposite

orientation than the aforementioned salient block, hence the number of columns of

height λk in this case can only be odd. This gives condition (D1).

Condition (D3) follows again from the absence of label 1 salient blocks. To see

this recall that in the bottom row of the type D pattern there are half blocks which

have label 0 in the odd columns and have label 1 in the even columns. Since there

are no salient blocks of label 1 in Y , this total number of columns can only be odd if

the last column has height 1, the column to the left of it has height 2, and so on.

This is can only happen when in the bottom row s = n− 2.

The fact that there is no salient block of label 2, . . . , n− 2 implies that for each

bead on the rulers R1, . . . , R2n−1 there has to be a block placed on its right. The

only exception is the ruler Rn−2. There cannot be any bead on this ruler, except

when there is a salient block pair of label n− 1/n which corresponds to a hole on

Rn−1. These considerations imply conditions (D4) and (D5).

Condition (D6) corresponds directly to property (R3). �

Given Y ∈ Z0∆, let ti denote the total number of beads in the i-th row of its

abacus representation, and li the number of beads in the rightmost position of the

i-th row. We obtain a sequence of pairs (ti, li)i∈Z+ , only finitely many of which do

not equal (0, 0).

Corollary 9.7. Given Y ∈ Z0∆, the associated sequence of pairs (ti, li)i∈Z+

satisfies the following conditions.

(1) For all i, 0 ≤ li ≤ ti.

(2) For all i, if ti > 0, then either li = ti is even, or li is odd.

(3) Either∑

i ti is even, or t1 − l1 = 2n− 4.

Conversely, any sequence (ti, li)i∈Z+ satisfying these conditions arises as a sequence

associated to at least one Young wall Y ∈ Z0∆. More precisely, the number of different


Young walls Y ∈ Z0∆ corresponding to any given sequence is 2m, where m is the

number of indices i such that ti − li > 2n− 2. All Young walls Y corresponding to a

single sequence have the same multi-weight, when the weights for labels n− 1 and n

are counted together.

Proof. Condition (1) is clear from the definition of (ti, li). Condition (2) follows

from (D1) above. Condition (3) is equivalent to condition (D3).

Conversely, given a sequence (ti, li)i∈Z+ satisfying conditions (1)-(3), we can

reconstruct a corresponding Y ∈ Z0∆ in its abacus representation as follows. On the

i-th row, li beads have to be put on the last position; (D1) is satisfied because of (1).

They are white if∑

j<i tj +∑

j>i,tj 6≡0 mod n−1 tj ≡ 1 mod 2, and black otherwise; this

is just a reformulation of (D2). (D3) is satisfied because of (3). At most one bead

can be put on each ruler between Rn and R2n−3, pushed to the right as much as

possible; this is (D4). If ti − li ≤ 2n− 2, then the rest of the beads fill up the rulers

between R1 and Rn−2, pushed to the right. If ti − li > 2n − 2, then there are no

beads in this row on the rulers between R1 and Rn−2; instead, the remaining beads

are all placed on the (n− 1)-st ruler, and they can be either white or black. These

rules reconstruct a configuration satisfying (D5)-(D6), and give the stated ambiguity

in the reconstruction. �

9.5. The generating series of distinguished 0-generated walls

In light of Theorem 8.14, in order to complete the proof of our main Theorem 3.5

for type D, we need the following combinatorial result, the precise analogue of

Proposition 4.19 in type A.

Theorem 9.8. Let ∆ be of type Dn, and let ξ be a primitive (2n− 1)-st root of

unity. Then the generating series of the set Z0∆ of distinguished 0-generated Young

walls is given in terms of the generating function of all Young walls by the following

substitution: ∑Y ∈Z0

∆

qwt0(Y ) = Z∆(q0, . . . , qn)∣∣∣q0=ξ2q, q1=···=qn=ξ

.

In analogy once again with the type A proof, the following is the key construction

in our proof of this result. There is a combinatorial map

p : Z∆ → Z0∆

defined as follows. For an arbitrary Young wall Y ∈ Z∆ we take the Young wall

Y1 ∈ Z ′∆ which contains Y and which is minimal with this property with respect

to containment. By Lemma 8.1 Y1 is unique. We let p(Y ) = red(Y1), where

red: Z ′∆ → Z0∆ is the map defined in Lemma 8.2. We remark that in fact p(Y ) is

the unique element in Z0∆ which has exactly the same set of label 0 blocks as Y .

9.5. THE GENERATING SERIES OF DISTINGUISHED 0-GENERATED WALLS 117

Proposition 9.9. On the abacus representation of Young walls, the map

p : Z∆ → Z0∆ corresponds to the following steps:

(1) If a row ends with a white (resp., black) bead on R2n−2 corresponding to a

column of height λk, and k + nk is even (resp. odd), then one bead should

be moved to the next position, which is the leftmost in the next row. This

is applied also on the zeroth position of the abacus, where we assume that

there are infinitely many beads. This corresponds to the appearance of a new

column in the Young wall represented by the abacus.

(2) Every bead on the rulers R1, . . . , R2n−4 is moved to right as much as possible

according to the abacus rules.

(3) If there is at least one bead on the rulers R1, . . . , R2n−3 after performing Step

1 on the previous row, and the number of beads on R2n−2 is even, then one

more bead is moved onto R2n−2. If there were beads on R2n−2 already, then

this step does not change the parity of k+nk for the rightmost bead. If there

were no beads on R2n−2 before, then this beads should take the appropriate

color and it is possible to see that it can not be moved further with Step 1.

(4) Let s be the total number of beads on the rulers R1, . . . , Rn−1 after performing

the Steps 1-3. If s > n− 2, then move all these beads on Rn−1. In this case

some of these beads were here previously, so the color of the whole group of

beads is already determined. If s ≤ n− 2, then move them onto the rulers

R1, . . . , Rn−2, each as right as possible.

Proof. Step 1 enforces condition (D2). It also enforces condition (D3) when

applied to the minus first row. Step 2 enforces conditions (D4) and (D5), Step 3

enforces condition (D1), and finally Step 4 enforces condition (D6). �

The fibers of the map p can be described as follows. Given a Young wall Y ∈ Z0∆,

we are allowed to move beads to the left and, occasionally, to the right, using the

following rules.

(1) From the last position of the i− th row only one (resp. zero) bead can be

moved to the left if li is odd (resp., even).

(2) Every other bead is allowed to moved to the left in its row if the result is a

valid abacus configuration.

(3) The leftmost bead in a row can be moved to the last position of the previous

row. There it will take the color white if∑

j<i tj +∑

j>i,tj 6≡0 mod n−1 tj ≡1 mod 2, and grey otherwise.

(4) If ti − li ≤ 2n− 2, then the beads to the left from the n− 1-st position are

allowed to be moved to the right at most onto the ruler Rn−1.

(5) If ti − li ≤ 2n − 2, then any configuration, in which there is at least one

bead at the (n− 1)-st position, has to be counted with multiplicity two.


Let us call the beads that can be moved according to these rules movable. For

a row with data (t, l), let us also introduce the number c(t, l), which is signed sum

of the distance of the beads from the Rn−1-st ruler, where the sum runs over the

movable beads, and the beads to the left of the Rn−1-st ruler are counted with

negative sign, and the beads to the right of it are counted with positive sign. These

numbers are listed in the table below:

l ≡ 0 mod 2 l ≡ 1 mod 2

0 ≤ t− l ≤ n− 2(n−1

2

)−(n−1−t+l

2

) (n2

)−(n−1−t+l

2

)n− 1 ≤ t− l ≤ 2n− 3

(n−1

2

)−(t−l−n+1

2

) (n2

)−(t−l−n+1

2

)2n− 2 ≤ t− l

(n−1

2

) (n2

)Lemma 9.10. The contribution of a row with data (t, l) to the total weight of the

fiber is ξ−c(t,l).

Proof. If l is even but nonzero, then according to Corollary 9.7 l = t and there

isn’t any movable bead.

If l is odd, then there is one movable bead on the R2n−2-nd ruler. Assume first

that t ≤ n− 1. Then the expression

2n−t+l−2∑n1=0

n1∑n2=0

· · ·nt−l∑

nt−l+1=0

(ξ−1)n1+···+nt−l+1 =

(2n− 1

t− l + 1

)ξ−1

counts once every preimage, in which there is at most one bead at the n − 1-st

position. Similarly

(ξ−1)n−t+l−1

2n−t∑n2=0

n2∑n3=0

· · ·nt−l∑

nt−l+1=0

(ξ−1)n2+···+nt−l+1 = (ξ−1)n−t+l−1

(2n− 1

t− l

)ξ−1

counts once every preimage, in which there is at least one and at most two beads at

the n− 1-st position, since we moved one bead from the leftmost occupied position

to the n− 1-st position, and we fixed it there. The next term is given by

(ξ−1)(n−t+l−1)+(n−t+l)2n−t∑n3=0

n3∑n4=0

· · ·nt−l∑

nt−l+1=0

(ξ−1)n3+···+nt−l+1

= (ξ−1)(n−t+l−1)+(n−t+l)(

2n− 1

t− l − 1

)ξ−1

,

which counts once every preimage, in which there is at least two and at most three

beads at the n− 1-st ruler. Continuing in this fashion and summing up in the end

we get (2n+ 1

t− l + 1

)ξ−1

+t−l+1∑i=1

ξ−∑n−t+l−2+ij=n−t+l−1 j

(2n− 1

t− l + 1− i

)ξ−1

.


It can be checked that(

2n−1k

)ξ−1 = 0 unless k = 0, in which case it is equal to 1.

Therefore, the only surviving part is the one with i = t− l + 1:

ξ−∑n−1j=n−t+l−1 j = ξ(

n2)−(n−1−t+l

2 ) = ξ−c(t,l).

The proofs of the remaining two cases, when l is odd, are very similar. The only

difference in the case 2n− 2 ≤ t− l is that first the preimages with zero or one extra

movable beads at Rn−1 have to be counted, then the preimages with two or three

extra movable beads, etc. As in the case t ≤ n− 1, the only nonzero term is the one

where in the beginning all the movable beads have been shifted to the Rn−1-st ruler,

and in this case the powers of ξ−1 sum up to(n2

)= c(t, l). �

Corollary 9.11. Let Y ∈ Z0∆ be a distinguished 0-generated Young wall de-

scribed by the data {(ti, li)i}. Then∑Y ′∈p−1

∗ (Y )

qwt(Y ′)∣∣∣q1=···=qn=ξ, q0=ξ2q

= qwt(Y )∣∣∣q1=···=qn=ξ, q0=ξ−(2n−3)q

· ξ−∑i c(ti,li)

= qwt0(Y )ξ∑j 6=0(wtj(Y )−dim ρj ·wt0(Y ))−

∑i c(ti,li)

Lemma 9.12. The core of a 0-generated Young wall is a 0-generated Young wall.

Proof. With each reduction step (B1)-(B4) we always remove a bar. In the

original Young wall, the salient blocks were only label 0 half blocks and salient

block-pairs of label n− 1/n.

A similar reasoning as in the type A case shows that no salient block of label

2, . . . , n− 2 can appear after we perform the step (B1) until possible. The same is

true with (B2), since if we can perform it on a pair of beads in a row, then we can

always perform it on the beads between them. More precisely, it can be seen that

even label 1 salient blocks cannot appear during these two steps because the parity

conditions in (D1) and (D2) is always maintained by the reduction steps.

The parity conditions in (D1) and (D2) are maintained by the step (B3) as well.

If we perform (B4) on a pair of beads, then we can perform it on this pair as long as

they disappear from the abacus. Hence, when performing (B4) until possible we also

get back the good parities. After the reduction is completed there cannot be any

bead on the rulers R1, . . . , Rn−2, and all the beads on the rulers Rn, . . . , R2n−3 are

right-adjusted. Therefore the conditions (D4) and (D5) are also satisfied.

If the total number of beads was initially even, then since no label 1 salient block

can appear, the total number of beads in the end must be even as well. So the final

Young wall will satisfy (D3). If in the total number of beads in the initial abacus

configuration is odd, then in the first row t1− l1 = 2n− 4. Hence, the beads on ruler

R2n−2 in the first row are necessarily white and one of them can be taken away from

the abacus with the step (B3), together with the beads on the other rulers using

(B2). This is an odd number of beads removed from the abacus. After this the total

number of beads is even, so we reduced to the earlier case. �


Proof of Theorem 9.8. In light of Corollary 9.11, it remains to show that


∑i c(ti,li) = 1.

We follow in the line of the proof the An case.

Step 1: Reduction to 0-generated cores. According to Lemma 9.12 the core of a

0-generated Young wall is a 0-generated core. It is immediate from the definition of

c(t, l) that the steps (B1), (B2) and (B4) leave the sum∑

i c(ti, li) unchanged, while

(B3) is a bit more complicated. Indeed,

• (B1) removes one movable bead from a row, and adds one to another on

the same ruler;

• (B2) removes two movable beads, but these two contribute with opposite

signs into c(t, l);

• (B4) either moves non-movable beads from R2n−2 onto Rn−1, or beads from

Rn−1 into non-movable beads on R2n−2.

Moving beads on Rn−1 according to (B3) does not affect the numbers c(t, l). If (B3)

moves a bead on the ruler R2n−2 between rows having l of different parity, then the

sum of the movable beads at the (2n− 2)-nd positions of the two rows is constant,

so∑

i c(ti, li) does not change after such a step. If (B3) were able to move a bead on

R2n−2 between rows both having odd l’s, then this can happen only if they have the

same color and if there is no bead on Rn−1 between them. This means that there

must be an even number of beads between them and they should have same kind of

top block, or odd number of beads and different kind of top blocks. Both cases are

forbidden in 0-generated Young walls due to Lemma 9.6. For the same reasons (B3)

cannot move beads on R2n−2 between rows both having even l’s.

It follows from these considerations that for all Y ∈ Z0∆


∑i c(ti,li) = ξ

∑j 6=0(wtj(core(Y ))−dim ρj ·wt0(core(Y ))−

∑i c(ti,li).

Step 2: Reduction to distinguished 0-generated cores.

As described above, for any 0-generated core Y there is a decomposition as λ(Y ) =

µ(Y ) ∪ ν(Y ), where µ(Y ) ∈ C1, ν(Y ) ∈ C2, and we have ν(Y ) = ν(0)(Y ) + ν(1)(Y ),

where ν(0)(Y ) and ν(1)(Y ) are two-cores, and parts in ν(1)(Y ) have colors given by

their parity. Since Y is 0-generated, the largest part of ν(Y ) has to be even, otherwise

there is no bead at the rightmost position in the last row of the abacus. Therefore,

the abacus represenation of Y can be described as follows.

(1) There is no bead on the rulers Rk for 1 ≤ k ≤ n− 2;

(2) On the ruler Rn−1 all the beads are of the same color, the beads are at the

first, let’s say, m positions, exactly one bead at each.

(3) The positions in the first m rows of the rulers Rk for n ≤ k ≤ 2n − 3 are

all filled up with beads, the other beads are pushed to the right as much as

possible.


(4) There is at least m beads on the ruler R2n−2, one at each of the first positions

and there is no space between them. The first m of these are all white. The

total number of the rest is even, and half of them is white, half of them is

black.

The abacus of a typical 0-generated core looks like this:

R1. . . Rn−2 Rn−1 Rn

. . . R2n−3R2n−2

. . .

......

......

. . .

It is not true that the core of a distinguished 0-generated Young wall is always

distinguished. But we can reduce each non-distinguished core further using the

reduction map red and then taking the core of the result using the steps (B1)-

(B4) again. The result of this is a distinguished 0-generated core. The first step

corresponds to shifting the beads on Rn−1 one step left. This decreases∑

i c(ti, li)

by m, and decreases∑

j 6=0(wtj(Y ) − dim ρj · wt0(Y )) also by m. The second step

does not change these numbers by the considerations of Step 1. So we are done once

we know the statement for distinguished 0-generated cores.

We remark here that during the first step the color of every second bead on the

ruler R2n−2 changes. In the end the color of the beads on this ruler will alternate.

Step 3: The case of distinguished 0-generated cores.

Let (z1, . . . , zn) be the integer tuple assigned to Y as in 9.3. Then zn−1 = znsince there is no bead on Rn−1 and the color of the beads on R2n−2 alternates. In

particular, zn−1 + zn is an even number.

We claim thatn∑

1=1

mn =∑i

c(ti, li).

Indeed, by Theorem 9.3 (3),∑n

1=1mn = −∑n−2

i=1 (n− 1− i)zi− (n− 1)c(zn−1 + zn)−(n− 1)b. The number of movable beads on R2n−2−i is −zi if 1 ≤ i ≤ n− 1, and it is

−c(zn−1 + zn)− b if i = 0. This proves the claim.

By Theorem 9.3 (1), for a core

qwt(Y ) = qm11 · · · · · qmnn (q1/2)m

>·C·m.


As in the type A case, on the right hand side of this expression q0 only appears in

the q1/2-term. Hence1

2m> · C ·m = wt0(Y ),

and

(q1/2)m>·C·m

∣∣∣q1=···=qn=ξ, q0=ξ2q

= qwt0(Y ).

On the other hand,

qwt(Y )∣∣∣q1=···=qn=ξ, q0=ξ2q

= qwt0(Y )ξ∑j 6=0(wtj(Y )−dim ρj ·wt0(Y )).

Therefore

ξ∑j 6=0(wtj(Y )−dim ρj ·wt0(Y )) = ξ

∑ni=1 mn = ξ

∑i c(ti,li),

and so indeed


∑i c(ti,li) = 1.

�

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http://webpages.math.luc.edu/~ptingley/

Summary

The punctual Hilbert scheme parameterizing the zero-dimensional subschemes of

a quasi-projective variety contains a large amount of information about the geometry

and topology of the base variety. The Hilbert schemes of points on smooth curves and

surfaces have been investigated for a long time. In the recent years a new direction

has emerged, which also allows singularities on the base variety. The aim of this

thesis is to describe the Euler characteristics of the Hilbert schemes parameterizing

the zero-dimensional subschemes of some basic classes of surface singularities.

The well-known simple singularities are the simplest type of normal surface

singularities, and it is known that they have an orbifold structure. There are at

least two natural version of the punctual Hilbert scheme in the case of quotient

singularities.

We study the geometry and topology of Hilbert schemes of points on the orb-

ifold surface [C2/G], respectively the singular quotient surface C2/G, where G is

a finite subgroup of SL(2, C) of type A or D. We give a decomposition of the

(equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a

certain combinatorial set, the set of Young walls. The generating series of Euler

characteristics of Hilbert schemes of points of the singular surface of type A or D

is computed in terms of an explicit formula involving a specialized character of the

basic representation of the corresponding affine Lie algebra; we conjecture that the

same result holds also in type E. Our results are consistent with known results

for type A, and are new for type D. The crystal basis theory of the fundamental

representation of the affine Lie algebra corresponding to the surface singularity (via

the McKay correspondence) plays an important role in our approach. The result

gives a generalization of Gottsche’s formula and has interesting modular properties

related to the S-duality conjecture.

The moduli space of torsion free sheaves on surfaces are higher rank analogs of

the Hilbert schemes. In type A our results reveal their Euler characteristic generating

function as well. Another very interesting class of normal surface singularities is the

so-called cyclic quotient singularities of type (p, 1). As an outlook we also obtain

some results about the associated generating functions.

Osszefoglalo

Egy kvaziprojektıv varietas nulla dimenzios reszsemait parameterezo pontozott

Hilbert sema nagy mennyisegu informaciot tartalmaz az alapvarietas geometriajarol

es topologiajarol. Sima gorbek es feluletek pontjainak Hilbert semait regota

vizsgaljak. Az utobbi evekben egy uj iranyzat is megjelent, miszerint az alap-

varietason szingularitasok is megengedettek. A disszertacio celja, hogy leırja nehany

alap feluletszingularitas osztaly eseten a nulla dimenzios reszsemakat parameterezo

Hilbert-semak Euler-karakterisztikait.

Az egyszeru szingularitasok a legegyszerubbek a normalis feluletszingularitasok

kozott, es ismert hogy orbifold strukturaval is rendelkeznek. Hanyadosszingularitasok

eseten legalabb ket termeszetes verzioja letezik a pontozott Hilbert-semanak.

Megvizsgaljuk a [C2/G] orbifold felulet, es a C2/G hanyadosfelulet pontjainak

Hilbert-semajat, ahol G az SL(2, C)-nek A vagy D tıpusu reszcsoportja. Az ek-

vivarians Hilbert-semaknak megadjuk egy felbontasat affin terekre, amiket Young

falaknak egy meghatarozott kombinatorikusan leırhato halmaza indexel. Explicit

alakban kiszamoljuk az A es D tıpusu szingularis feluletek Hilbert-semainak Euler-

karakterisztikainak generatorfuggvenyet is, ami egy specializacioja a megfelelo affin

Lie-algebra alapreprezentaciojanak karakterenek. Az E tıpus esetet sejteskent fogal-

mazzuk meg. Az eredmenyeink konzisztensek a korabbiakkal az A tıpusra, es ujak a

D tıpusra. A megkozelıtesunkben fontos szerepet jatszik az feluletszingularitashoz a

McKay-korrespondencian keresztul tartozo affin Lie-algebra alapreprezentaciojanak

kristalybazis elmelete. Az eredmenyek altalanosıtjak Gottsche formulajat es az

S-dualitas sejteshez kapcsolodoan erdekes modularitasi tulajdonsagokkal is ren-

delkeznek.

A torziomentes kevek modulustere a pontozott Hilbert sema egy magasabb

dimenzios analogiaja. Az A tıpus eseteben az itt ismertetett modszer megadja

ezen modulusterek Euler-karakterisztikainak generatorfuggvenyet is. A normalis

feluletszingularitasok egy masik, nagyon erdekes osztalya a (p, 1)-tıpusu ciklikus

szingularitasoke. Kitekinteskent az ezekhez asszocialt generatorfuggvenyekrol is

bemutatunk nehany eredmenyt.

hilbert schemes of points on some classes of surface singularitiesagyenge/thesis_phd.pdf · hilbert...

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